Question

Given that a randomly chosen card from a standard deck of 52 cards is less than 7, what is the probability it is the 5 of diamonds? Assume that aces are low cards.

Answer 1

Solution

The picture below shows the whole 52 standard deck

Let B denotes the events of cards that is less than 7

We will include the aces in the set B, because we are told to assume that they are low cards

B = {all 2, all 3, all 4, all 5, all 6, all aces}

n(B) = 24

Let A denotes the events of cards that is a 5 of diamonds

A = {5diamond}

n(A) = 1

A n B = {5diamonds}

n(A n B) = 1

The probability

$\begin{gathered} p(A\cap B)=\frac{1}{52} \\ p(B)=\frac{24}{52} \end{gathered}$

Note: Conditional Probability Formula

$p(A|B)=\frac{p(A\cap B)}{p(B)}$

From the question, we want to find the probability of A given B

$\begin{gathered} p(A|B)=\frac{p(A\cap B)}{p(B)} \\ p(A|B)=\frac{\frac{1}{52}}{\frac{24}{52}} \\ p(A|B)=\frac{1}{52}\times\frac{52}{24} \\ p(A|B)=\frac{1}{24} \end{gathered}$

Therefore, the answer is

$\frac{1}{24}$