Logic (Hurley, Chapter 11, "which Probability Rule Do I Use") Joy

Front 1

Hat One contains 6 yellow marbles and 4 red marbles. Hat Two contains 3 yellow marbles and 3 red marbles and 4 green marbles. If I draw one marble from Hat One and one marble from Hat Two, what is the probability that I have drawn two yellow marbles? Which rule do I use?

Back 1

Goal: Y and Y.
Neither draw affects the other.
Restricted conjunction rule.
6/10 x 3/10

Front 2

What is the probability of getting a heads on either (or both) of two tosses of a coin? What rule do I use?

Back 2

Goal: H on coin 1 or on coin 2 (or both).
Not mutually exclusive (since obviously getting a H on one coin does not mean one cannot get H on the other.)
General disjunction rule.
1/2 + 1/2 - (1/2 x 1/2) = 3/4

Front 3

If the probability that a person is female is .51 and the probability that a person is red-headed is .14, what is the probability that the next person you randomly encounter will be a red-headed female? What rule do I use?

Back 3

(a red-headed female is both a red-head and female)
Goal: red-headed and female.
Restricted conjunction rule.
.51 x .14

Front 4

What is the probability of getting at least one six when rolling a pair of dice? [This question not about adding the two dice together.] What rule do I use?

Back 4

Goal: a 6 on one die or on the other (or both).
Not mutually exclusive.
General disjunction rule.
P(6 on 1st die) + P(6 on 2nd die) - [P(6 on 1st) x P(6 on 2nd)]
= 1/6 + 1/6 - [1/6 x 1/6]

Front 5

What is the probability of getting two aces from a deck of cards on two draws without replacement? What rule do I use?

Back 5

Goal: Ace on first draw and ace on second draw.
The first draw affects the second draw.
General conjunction rule.
P(A1 and A2) = 4/52 x 3/51

Front 6

What is the probability of drawing either a nine or a King from a deck on a single draw? What rule do I use?

Back 6

Goal: either a 9 or a K.
Only one draw so the events are mutually exclusive (getting 9 precludes getting K and vice versa.
Restricted disjunction rule.
P(9 or K) = 4/52 + 4/52 = 8/52

Front 7

A concert hall has ten rows of ten seats. In a random drawing for seats you are the first to draw two tickets. What is the probability of drawing two front row seats? What rule do I use?

Back 7

Goal: Front row seat on 1st draw and front row on 2nd draw.
First draw affects the second.
General conjunction rule.
Pr(F1 and (F2 given F1))
= 10/100 x 9/99 = 1/110

Front 8

What is the probability of getting either a three or an even number when rolling a die? What rule do I use?

Back 8

Goal: 3 or even number.
It's only one roll of one die so the possibilities are mutually exclusive.
P(3 or E) = 1/6 + 3/6 = 4/6

Front 9

Hat One contains 6 yellow marbles and 4 red marbles. Hat Two contains 3 yellow marbles and 3 red marbles and 4 green marbles. What if I draw one marble from Hat One and put it in my pocket, and then draw another marble from the same hat (Hat One) and put it in my pocket, and then draw a third marble from the same hat (Hat One). What is the probability that I have three yellow marbles in my pocket? Which rule do I use?

Back 9

Goal: Yellow and yellow and yellow
Each draw affects the next draw.
General conjunction rule.
6/10 x 5/9 x 4/ 8

Front 10

A 10-sided die (numbered 1-10) is rolled. What is the probability of getting 4 or 8? What rule do I use?

Back 10

Goal: 4 or 8.
One roll so events are mutually exclusive (can't both happen).
Restricted disjunction rule.
1/10 + 1/10

Front 11

What is the probability of drawing a Jack or a king from a standard deck (with one draw)? What rule do I use?

Back 11

Goal: J or K.
Mutually exclusive since it is only one draw.
Restricted disjunction rule.
4/52 + 4/52 = 8/52 = 2/13

Front 12

Hat One contains 6 yellow marbles and 4 red marbles. Hat Two contains 3 yellow marbles and 3 red marbles and 4 green marbles. What if I draw one marble from Hat Two and put it in my pocket, and then draw another marble from the same hat (Hat Two) and put it in my pocket. What is the probability that I have two green marbles in my pocket? Which rule do I use?

Back 12

Goal: G and G.
Same hat, so the first draw affects the second.
General conjunction rule.
4/10 x 3/9

Front 13

Hat One contains 6 yellow marbles and 4 red marbles. Hat Two contains 3 yellow marbles and 3 red marbles and 4 green marbles. What is the probability of drawing either a yellow marble or a red marble on a single draw from Hat One? Which rule?

Back 13

Goal: Y or R.
Single draw, so if one gets Y he cannot get R and vice versa. (Mutually exclusive)
Restricted disjunction rule.
6/10 + 4/10

Front 14

What's the probability of drawing a jack or a spade? What rule do I use?

Back 14

Goal: J or spade.
Not mutually since there is a jack of spades in the deck.
General disjunction rule.
So, it is the P of a jack plus the P of a spade minus the P of the jack of spades (1/52).
4/52 + 13/ 52 - 1/52 = 16/52

Front 15

If five of 50 tires in a tire store are in some way defective, what is the probability that four tires put on your car will all be defective? What rule do I use?

Back 15

"Goal": D and D and D and D.
Not independent since each tire selected reduces the inventory.
General conjunction rule.
5/50 x 4/49 x 3/48 x 2/47
= 1/46060

Front 16

In Clarajean's closet are four pairs of pants (black, white, grey, and brown), and five different shirts (blue, white, red, yellow, and purple). If she was to blindly pick a pair of pants and a shirt, what is the probability that the pants and shirt will be the same color?

Back 16

Goal: same color pants and shirt = White and White.
Independent events since which pants are chosen does not affect which shirt is chosen and vice versa.
Restricted conjunction rule.
1/4 x 1/5 = 1/20 = 0.05 = 5%

Front 17

Hat One contains 6 yellow marbles and 4 red marbles. Hat Two contains 3 yellow marbles and 3 red marbles and 4 green marbles. What if I draw one marble from Hat One and put it in my pocket, and then draw another marble from the same hat (Hat One) and put it in my pocket. What is the probability that I have two red marbles in my pocket? Which rule do I use?

Back 17

Goal: R and R.
The first draw affects the second.
General conjunction rule.
4/10 x 3/9

Front 18

A 10-sided die (numbered 1-10) is rolled. What is the probability of getting 9 or a number less than three? What rule do I use?

Back 18

Goal: 9 or (1 or 2).
One roll, so events are mutually exclusive (if a 1 is rolled, the other possibilities cannot occur).
Restricted disjunction rule.
1/10 + 1/10 + 1/10 = 3/10

Front 19

Hat One contains 6 yellow marbles and 4 red marbles. Hat Two contains 3 yellow marbles and 3 red marbles and 4 green marbles. What is the probability of drawing either a yellow marble or a red marble on a single draw from Hat Two? What rule do I use?

Back 19

Goal: R or Y
Mutually exclusive: A single draw so if one gets R they cannot get Y and vice versa.
Restricted disjunction rule.
3/10 + 3/10

Front 20

Hat One contains 6 yellow marbles and 4 red marbles. Hat Two contains 3 yellow marbles and 3 red marbles and 4 green marbles. What is the probability of drawing either a green marble or a red marble on a single draw from Hat Two? Which rule do I use?

Back 20

Goal: G or R
Single draw, so if one gets G she cannot get Y and vice versa. (Mutually exclusive)
Restricted disjunction rule.
4/10 + 3/10

Front 21

A 10-sided die (numbered 1-10) is rolled. What is the probability of getting 5 or an odd number? What rule do I use?

Back 21

Goal: 5 or (1 or 3 or 5 or 7 or 9).
Not mutually exclusive since 5 is both a 5 and odd (notice it is listed twice).
General disjunction rule.
1/10 + 5/10 - 1/10 = 5/10
= 1/2 = 0.5 = 50%

Front 22

Roll a regular 6-sided die. Find the probability of rolling a 2 or an even number. What rule do I use?

Back 22

Goal: 2 or (2 or 4 or 6).
These events are inclusive (not mutually exclusive) because a 2 and an even number can happen at the same time. A 2 is an even number, so this would satisfy both.
General disjunction rule.
P(A or B) = P(A) + P(B) - P(A and B)
1/6 + 3/6 - 1/6 = 3/6 = 1/2

Front 23

What is the probability of getting heads on each of three successive tosses of one coin? What rule do I use?

Back 23

Goal: H and H and H.
Restricted conjunction rule.
1/2 x 1/2 x 1/2 = 1/8

Front 24

What is the probability of getting at least one six on three rolls of a die?
How would I set this up to use the negation rule?

Back 24

Negation rule: 1 - (prob. of NOT getting at least one 6.
1 - (prob. of getting non-6 and non-6 and non-6).
1 - (5/6 x 5/6 x 5/6)

Front 25

What is the probability of getting at least one tail on three tosses of a coin? How would I set this up to use the negation rule?

Back 25

Negation rule: 1 - (prob. of NOT getting at least one Tail).
1 - (getting all heads) = 1 - (H and H and H)
1 - (1/2 x 1/2 x 1/2) = 1 - 1/8 = 7/8

Front 26

Assume that a person can have only one of four possible hair colors and that the probabilities of each are: black (.31); brown (.30); blond (.25); or red (deliberately unspecified). Assume that no one is bald. Assume that every person is either male or female, that sex and hair color are independent of one another, and that the probability of being female is .51. What is the probability of a red-haired male? What rule would I use?

Back 26

Goal: red-haired male (a person both red-haired and male).
Restricted conjunction rule. (And using the negation rule to get the probabilities that one is red-haired and that one is male).
[ 1 - (.31 + .30 + .25)] x (1 - .51)
= .14 x .49

Front 27

Hat One contains 6 yellow marbles and 4 red marbles. Hat Two contains 3 yellow marbles and 3 red marbles and 4 green marbles. What if I draw one marble from Hat Two and put it in my pocket, and then draw another marble from the same hat (Hat Two) and put it in my pocket, and then draw a third marble from the same hat (Hat Two). What is the probability that I have three red marbles in my pocket? What rule do I use?

Back 27

Goal: R and R and R.
Each draw affects the next.
General conjunction rule.
3/10 x 2/9 x 1/8

Front 28

Captain Omar and his ship, the H.M.S. Khan, are two furlongs from the dread pirate Ishaan and his merciless band of scallawags. The Captain has probability 1/2 of hitting the pirate ship. The pirate only has one good eye, so he hits the Captain's ship with probability 2/5.
If both fire their cannons at the same time, what is the probability that the pirate hits the Captain's ship, but the Captain misses? What rule do I use?

Back 28

Goal: pirate hits and the Captain misses.
Independent events. Use negation rule to figure out chance of Captain missing if he has a 1/2 chance of hitting.
2/5 x (1 - 1/2) = 2/5 x 1/2 = 2/10 = 0.2 = 20%

Front 29

If a pair of dice are rolled, what is the probability that the total points (pips, dots) add up to five? What rule do I use?

Back 29

Goal: (1 and 4) or (2 and 3) or (3 and 2) or (4 and 1).
Use conjunction to figure each way of getting a five; use restricted disjunction to calculate the total probability.
(1/6x1/6) + (1/6x1/6) + (1/6x1/6) + (1/6x1/6) = 4/36

Front 30

Hat One contains 6 yellow marbles and 4 red marbles. Hat Two contains 3 yellow marbles and 3 red marbles and 4 green marbles. What is the probability that drawing one marble from each hat will yield two red marbles? Which rule do I use?

Back 30

Goal: Red and Red
Neither draw affects the other (independent events).
Restricted conjunction rule.
4/10 x 3/10

Front 31

LeBron Asks: What are the chances of making 10 free throws in a row? Assume the probability of making each individual shot is 75%. What rule do I use?

Back 31

Goal: Make the first shot and make the second and ....
Restricted conjunction rule.
See the Kahn acadamy video: https://www.khanacademy.org/math/probability/independent-dependent-probability/independent_events/v/lebron-asks-about-the-chances-of-making-10-free-throws

Front 32

Assume that a person can have only one of four possible hair colors and that the probabilities of each are: black (.31); brown (.30); blond (.25); or red (deliberately unspecified). Assume that no one is bald. Assume that every person is either male or female, that sex and hair color are independent of one another, and that the probability of being female is .51. What is the probability of a person with either red or blond hair? What rule do I use?

Back 32

Goal: red hair or blond.
Restricted disjunction rule, but figure red hair by negation rule.
[1 - (.31 + .30 + .25)] + .25
= .14 + .25

Front 33

Tim Duncan is shooting free throws. Making or missing free throws doesn't change the probability that he will make his next one, and he makes his free throws 72% of the time.
What is the probability of Tim Duncan making ALL of his next 8 free throw attempts? What rule do I use?

Back 33

Goal: Make the basket and Make and Make and ....
Restricted conjunction rule.
.72 x .72 x .72 x .72 x .72 x .72 x .72 x .72

Front 34

If you flip a coin and roll a 6-sided die, what is the probability that you will flip a tails and roll less than a 6? What rule do I use?

Back 34

Goal: Tails and (either a 5 or 4 or 3 or 2 or 1).
Independent events so use restricted conjunction rule, but use the disjunction rule to add up the probabilities of tossing less than a 6.
1/2 X (1/6 + 1/6 + 1/6 + 1/6 +16) = 1/2 x 5/6 = 5/12

Front 35

Captain Emily and her ship, the H.M.S. Khan, are two furlongs from the dread pirate William and his merciless band of scallawags. The Captain has probability 3/8 of hitting the pirate ship. The pirate only has one good eye, so he hits the Captain's ship with probability 1/4.
If both fire their cannons at the same time, what is the probability that the Captain hits the pirate ship, but the pirate misses?

Back 35

Goal: Captain Emily hits and the pirate misses.
Restricted conjunction rule, but figure the probability that the pirate misses with the negation rule.
3/8 x (1 - 1/4) = 3/8 x 3/4 = 9/32