Independent and Dependent Events

Watch this lecture series to see more various examples of calculating probabilities of independent and dependent events. When the outcome of one event depends on the outcome of another, the events are considered dependent. Complete the interactive exercises

Practice

Interpret probabilities of compound events - Questions

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.

(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.

(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.

(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":