Question

a board game uses the deck of 20 cards shown to the right. two cards are selected at random from this deck. calculate the probability that both cards selected have a 5, both with and without replacement. two cards are to be selected with replacement. determine the probability that both cards selected have a 5. \\( \frac { 1 } { 25 } \\) (type an integer or a simplified fraction.) two cards are to be selected without replacement. determine the probability that both cards selected have a 5. (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate the number of cards with a 5

There are 4 cards with a 5 in the deck of 20 cards.

Step2: Probability without replacement

The probability of drawing a 5 on the first draw is $\frac{4}{20}$.
Since we are not replacing the card, for the second draw, there are now 3 cards with a 5 left out of 19 cards. So the probability of drawing a 5 on the second draw is $\frac{3}{19}$.
The probability of both events happening is the product of these two probabilities: $\frac{4}{20}\times\frac{3}{19}$.
Simplify $\frac{4}{20}$ to $\frac{1}{5}$. Then $\frac{1}{5}\times\frac{3}{19}=\frac{3}{95}$.

Answer:

$\frac{3}{95}$