# Conditional Probability Problems about Die Rolling

## Problem 728

A fair six-sided die is rolled.

(1) What is the conditional probability that the die lands on a prime number given the die lands on an odd number?

(2) What is the conditional probability that the die lands on 1 given the die lands on a prime number?

## Solution.

Let $E$ and $F$ be events. Then the conditional probability that $E$ occurs given $F$ has occurred, which is denoted by $P(E\mid F)$, is defined as
$P(E \mid F) = \frac{P(E \cap F)}{P(F)}.$

For uniform space (such as fair die rolling), this can be rewritten as
$P(E \mid F) = \frac{|E \cap F|}{|F|}.$ Here the notation $|A|$ means the number of elements in $A$.

### Solution (1)

Let $E = \{2, 3, 5\}$ be the set of prime number faces of a die. Let $F = \{1, 3, 5\}$ be the set of odd faces of a die. Then the required conditional probability can be calculated as follows:
\begin{align*}
P(E \mid F) &= \frac{|E \cap F|}{|F|}\\
&= \frac{|\{2, 3, 5\} \cap \{1, 3, 5\}|}{|\{1, 3, 5\}|}\\
&= \frac{|\{3, 5\}|}{\{1,3,5\}|}\\
&= \frac{2}{3}
\end{align*}

### Solution (2)

The required conditional probability is
\begin{align*}
P(\{1\} \mid E) &= \frac{|\{1\} \cap E|}{|E|}\\
&= \frac{|\{1\} \cap \{2, 3, 5\}|}{|\{2, 3, 5\}|}\\
&= \frac{|\emptyset|}{|\{2, 3, 5\}|}\\
&= \frac{0}{3} = 0
\end{align*}

### More from my site

• Probabilities of An Infinite Sequence of Die Rolling Consider an infinite series of events of rolling a fair six-sided die. Assume that each event is independent of each other. For each of the below, determine its probability. (1) At least one die lands on the face 5 in the first $n$ rolls. (2) Exactly $k$ dice land on the face 5 […]
• Probability Problems about Two Dice Two fair and distinguishable six-sided dice are rolled. (1) What is the probability that the sum of the upturned faces will equal $5$? (2) What is the probability that the outcome of the second die is strictly greater than the first die? Solution. The sample space $S$ is […]
• Overall Fraction of Defective Smartphones of Three Factories A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively. Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively. Determine the overall fraction of […]
• Lower and Upper Bounds of the Probability of the Intersection of Two Events Let $A, B$ be events with probabilities $P(A)=2/5$, $P(B)=5/6$, respectively. Find the best lower and upper bound of the probability $P(A \cap B)$ of the intersection $A \cap B$. Namely, find real numbers $a, b$ such that $a \leq P(A \cap B) \leq b$ and $P(A \cap B)$ could […]
• What is the Probability that All Coins Land Heads When Four Coins are Tossed If…? Four fair coins are tossed. (1) What is the probability that all coins land heads? (2) What is the probability that all coins land heads if the first coin is heads? (3) What is the probability that all coins land heads if at least one coin lands […]
• What is the Probability that Selected Coin was Two-Headed? There are three coins in a box. The first coin is two-headed. The second one is a fair coin. The third one is a biased coin that comes up heads $75\%$ of the time. When one of the three coins was picked at random from the box and tossed, it landed heads. What is the probability […]
• If At Least One of Two Coins Lands Heads, What is the Conditional Probability that the First Coin Lands Heads? Two fair coins are tossed. Given that at least one of them lands heads, what is the conditional probability that the first coin lands heads? We give two proofs. The first one uses Bays' theorem and the second one simply uses the definition of the conditional […]
• Independent Events of Playing Cards A card is chosen randomly from a deck of the standard 52 playing cards. Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart. Prove or disprove that the events $E$ and $F$ are independent. Definition of Independence Events […]

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

##### Probability Problems about Two Dice

Two fair and distinguishable six-sided dice are rolled. (1) What is the probability that the sum of the upturned faces...

Close