Exercise 2 Page 700

The probability that an event will occur, given that another event has already occurred, is called a conditional probability. There are $52$ cards in a standard deck of cards. There are $4$ different colored suits - red heart, red diamond, black club, and black spade. Each suit includes an Ace, a King, a Queen, a Jack and numerals from $1$ to $10.$

Since we want to know the conditional probability of selecting a card number $4$ from the deck. Given that the card drawn is black, let's first calculate the number of $\text{black cards}$ $\text{numbered }4.$

There is $1$ card numbered four in each suit and there are $2$ black suits, therefore there are $2$ black cards numbered four. Now, we can calculate total number of $\text{black cards}.$

$$\begin{array}{rlr}\text{black club }\text{cards: }& \frac{52}{4}=13& \\ \text{black spade }\text{cards: }& \frac{52}{4}=13& \\ \text{black }\text{cards: }& 13+13=26& \end{array}$$

The condition that the card selected is black limits the total outcome to $26$ cards. Out of those $26$ black cards, there are $2$ numbered four.

$P(4\text{ | Black})=\frac{\text{Card number four}}{\text{Black card}}$

SubstituteValues

Substitute values

$P(4\text{ | Black})=\frac{2}{26}$

ReduceFrac

$\frac{a}{b}=\frac{a/2}{b/2}$

$P(4\text{ | Black})=\frac{1}{13}$

▼

Convert to percent

CalcQuot

Calculate quotient

$P(4\text{ | Black})=0.076923\dots$

RoundDec

Round to $3$ decimal place(s)

$P(4\text{ | Black})\approx 0.077$

WritePercent

Convert to percent

$P(4\text{ | Black})\approx 7.7\%$

The probability of selecting a black diamond card is $\frac{1}{13},$ which can be also written as $7.7\%.$