Independent and Dependent Events

$\begin{array}{lccc}\text { Grade } & 9^{\text {th }} & 10^{\text {th }} & 11^{\text {th }} \\\hline \text { Number of students } & 12 & 9 & 7\end{array}$

Suppose that Ana picks a name, replaces it, and picks a name again.

What is the probability that NEITHER of the students selected are $9^{\text {th }}$ graders?

Round your answer to two decimal places.

_________

Suppose that $2$ bags go through the system, one after the other.

What is the probability that both bags are selected for extra screening?

Round your answer to two decimal places.

Hamid has no idea what the answers to a certain question are, so he needs to choose two different answers at random.

What is the probability that Hamid guesses both answers correctly?

Round your answer to two decimal places.

Sammy and Hannah both want to get leeks as their star ingredient. Sammy will draw first, followed by Hannah.

What is the probability that NEITHER contestant draws leeks?

Round your answer to two decimal places.

1. The probability that neither of the students selected are $9^{\text {th }}$ graders is about $0.33$.

2. The probability that both bags are selected for extra screening is about $0.02$.

3. The probability that Hamid guesses both answers correctly is $0.1$.

4. The probability that both contestants get the ingredient they want is about $0.67$.

1. Contestants on a gameshow spin a wheel with $24$ equally-sized segments. Most of those segments show different prize amounts, but $2$ of them are labeled "bankrupt":

Suppose that a contestant is going to spin the wheel twice in a row. Here are some events and their meanings:

$\begin{array}{cl}\hline \text { Event } & \text { Meaning } \\\hline B_{1} & \text { The first spin lands on bankrupt. } \\\hline B_{2} & \text { The second spin lands on bankrupt. } \\\hline B_{1}^{C} & \text { The first spin does not land on bankrupt. } \\\hline B_{2}^{C} & \text { The second spin does not land on bankrupt. } \\\hline \end{array}$

Consider this probability:

$P\left(B_{1}^{C} \text { and } B_{2}^{C}\right)=P\left(B_{1}^{C}\right) \cdot P\left(B_{2}^{C} \mid B_{1}^{C}\right)$

What does $P\left(B_{1}^{C}\right.$ and $\left.B_{2}^{C}\right)$ represent in this context?

Choose 1 answer:

(A) The probability that neither spin lands on bankrupt.

(B) The probability that both spins land on bankrupt.

(C) The probability that the first spin lands on bankrupt given the second spin lands on bankrupt.

(D) The probability that the second spin lands on bankrupt given the first spin lands on bankrupt.

2. A standard deck of 52 cards has 13 clubs, 13 diamonds, 13 hearts, and 13 spades.

Suppose that Ramón is going to draw 2 cards without replacement. Here are some events and their meanings:

$\begin{array}{cl}\text { Event } & \text { Meaning } \\\hline D_{1} & \text { The first card is a diamond. } \\\hline D_{2} & \text { The second card is a diamond. } \\\hline D_{1}^{C} & \text { The first card is not a diamond. } \\\hline D_{2}^{C} & \text { The second card is not a diamond. } \\\hline\end{array}$

Consider this probability:

$P\left(D_{1}^{C} \text { and } D_{2}^{C}\right)=P\left(D_{1}^{C}\right) \cdot P\left(D_{2}^{C} \mid D_{1}^{C}\right)$

What does $P\left(D_{1}^{C}\right.$ and $\left.D_{2}^{C}\right)$ represent in this context?

Choose 1 answer:

(A) The probability that neither card is a diamond.

(B) The probability that both cards are diamonds.

(C) The probability that the second card is a diamond given the first card is a diamond.

(D) The probability that the second card is a diamond given the first card is NOT a diamond.

3. Youssef is a basketball player who regularly shoots sets of 2 free-throws. Here are some events and their meanings:

$\begin{array}{cl}\hline  \text { Event } & \text { Meaning } \\\hline S_{1} & \text { He makes the first shot. } \\\hline S_{2} & \text { He makes the second shot. } \\\hline S_{1}^{C} & \text { He misses the first shot. } \\\hline  S_{2}^{C} & \text { He misses the second shot. } \\\hline \end{array}$

Suppose that Youssef is going to shoot 2 free-throws. Consider this probability:

$P\left(S_{1}^{C}\right.$ and $\left.S_{2}^{C}\right)=P\left(S_{1}^{C}\right) \cdot P\left(S_{2}^{C} \mid S_{1}^{C}\right)$

What does $P\left(S_{1}^{C}\right.$ and $\left.S_{2}^{C}\right)$ represent in this context?

Choose 1 answer:

(A) The probability that Youssef makes both shots.

(B) The probability that Youssef misses both shots.

(C) The probability that Youssef makes the second shot given he misses the first.

(D) The probability that Youssef misses the second shot given he makes the first.

4. Contestants on a gameshow spin a wheel with $24$ equally-sized segments. Most of those segments show different prize amounts, but $2$ of them are labeled "bankrupt":

Suppose that a contestant is going to spin the wheel twice in a row. Here are some events and their meanings:

$\begin{array}{cl}\text { Event } & \text { Meaning } \\\hline B_{1} & \text { The first spin lands on bankrupt. } \\B_{2} & \text { The second spin lands on bankrupt. } \\B_{1}^{C} & \text { The first spin does not land on bankrupt. } \\B_{2}^{C} & \text { The second spin does not land on bankrupt. }\end{array}$

Consider this probability:

$P\left(B_{1}\right.$ and $\left.B_{2}\right)=P\left(B_{1}\right) \cdot P\left(B_{2} \mid B_{1}\right)$

What does $P\left(B_{2} \mid B_{1}\right)$ represent in this context?

Choose 1 answer:

(A) The probability that neither spin lands on bankrupt.

(B) The probability that both spins land on bankrupt.

(C) The probability that the second spin lands on bankrupt given the first spin does NOT land on bankrupt.

(D) The probability that the second spin lands on bankrupt given the first spin lands on bankrupt.

1. $P\left(B_{1}^{C}\right.$ and $\left.B_{2}^{C}\right)$ represents the probability that neither spin lands on bankrupt.

2. $P\left(D_{1}^{C}\right.$ and $\left.D_{2}^{C}\right)$ represents the probability that neither card is a diamond.

3. $P\left(S_{1}^{C}\right.$ and $\left.S_{2}^{C}\right)$ represents the probability that Youssef misses both shots.

4. $P\left(B_{2} \mid B_{1}\right)$ represents the probability that the second spin lands on bankrupt given the first spin lands on bankrupt.