Deck 3: Conditional Probability and Independence

Question 1

Two friends are running a 5K together. If their finishing times are random in the range of 26 to 40 minutes, and they finish independently of one another, what is the probability that one finishes in under 32 minutes, while the other takes at least 37 minutes?

Answer 1

$\text { 2. } \frac { 6 } { 14 } \cdot \frac { 3 } { 14 }$

Question 2

A machine produces batches of 10 light-bulbs at a time. The probability any given light bulb will be defective is 15%. A batch is rejected if at least 3 light-bulbs are defective. Find the probability a given batch is rejected.

Answer 2

$1 - \left( \left( \begin{array} { c } 10 \\0\end{array} \right) ( .85 ) ^ { 10 } ( .15 ) ^ { 0 } + \left( \begin{array} { c } 10 \\1\end{array} \right) ( .85 ) ^ { 9 } ( .15 ) ^ { 1 } + \left( \begin{array} { c } 10 \\2\end{array} \right) ( .85 ) ^ { 8 } ( .15 ) ^ { 2 } \right) = .18$

Question 3

Your cell phone battery is getting quite low, and you know that it will turn off randomly between now and the next 50 minutes. You are talking with a dear friend. If you manage to talk for 30 minutes, what is the probability that your phone lasts at least 12 more minutes?

Answer 3

$\frac { 8 / 50 } { 20 / 50 } = \frac { 8 } { 20 } = .4$

Question 4

You have four $1 bills, three $5 bills, and a $10 bill. Your friend Abdullah has seven $1s, two $5s, and three $10s. You notice that one of you has dropped a bill on the ground. If it is a $5, what is the probability that it was your bill?

Question 5

Suppose that 3 fair four-sided dice are rolled, and we're told the sum of these is even. What is probability one of the rolls was a 3?

Question 6

For lunch, every day, two of your friends, Mary and Jenn either have pizza or salad. Mary eats pizza 5 days per week and salad twice. Jenn eats pizza once a week and salad 6 times. Referring to Mary and Jenn, on a certain day, you are told that one of your friends had pizza, but they cannot tell you which friend. What is the probability that it was Mary who was seen having pizza?

Question 7

You and your friend alternate drawing cards from a standard deck at random with replacement. The game ends when either you draw a red card, or he draws a face card (Jack, Queen, King). You draw first. What is the probability that you win (i.e., you draw a heart before he draws a face card)? You may leave your answer as an infinite sum.

Question 8

In genetics, blonde hair (b) is dominant over red hair (r). Gary has blonde hair, meaning he either has genetic pairing BB or Br. He is unsure of the probability of each outcome. His doctor tells him if he marries a woman with red hair (genetic marker rr) their child will be red haired with probability 3/8. What is the probability that Gary has genetic marker BB?

Question 9

Construct a sample space and two events A and B so that P(A|B)=2P(B|A).

Question 10

A box contains 15 quarters, one of which has heads on both sides. The remaining 14 are normal. You pick a quarter out of the box at random and flip it six times. All six times you get heads. What is the probability it is the 2-headed quarter?

Question 11

At a certain radio station, they receive requests all day via email. They play a mix of requested songs and songs chosen by the DJ; they always play 21% of songs requested by 9am, 31% of the songs requested between 9am and 2pm, and 8% of songs requested after 2pm. If in a given day 14% of requests are received by 9am and 43% are received by 2pm (including those by 9am), what percentage of songs played that day are chosen by the DJ?

Question 12

A baseball team has 4 pitchers and 4 catchers, all of whom play in a given game. The team manager records a defensive success if at least 2 of the pitchers and 3 of the catchers make no mistakes. In a given game, a pitcher makes no mistakes with probability .6 independently of any other player and a catcher makes no mistakes with probability .7. Give an exact expression for the probability that a randomly observed game is recorded as a success.

Question 13

A certain type of dog can have long hair, medium hair, or short hair. For any length of hair, the dog can be white, grey, yellow, or brindle. Let $E _ { L } , E _ { M } , E _ { H }$ be the events of long hair, medium hair, and short hair, respectively. Let $C _ { W } , C _ { G } , C _ { Y } , C _ { B }$ be the events that a dog is white, grey, yellow, or brindle respectively. Denote $D _ { a , b } = E _ { a } C _ { b } \text { for } a \in \{ L , M , H \} , b \in$ $\{ W , G , Y , B \}$ Give a partition of this breed of dogs into 6 sets.

Question 14

Urn I contains 3 black balls, 2 white balls, and a red ball. Urn II contains 5 black balls, 3 white balls, and 6 red balls. Urn III contains 4 black balls, 3 white balls, and 3 red balls. An urn is selected at random, one of the balls is removed randomly, observed to be white and not replaced. If another ball is then selected at random from the same urn, what is the probability it is black?

Question 15

At a certain stream, there are salmon and there are trout. Javon, Smitty, and Alice caught 22 fish and mixed them together; they have 14 salmon and 8 trout. Now, they can't remember who caught what. Javon knows that he caught 8 fish and the first 4 were trout. What is the probability that at least 2 of the last 4 he caught were Salmon?