Question

Stephanie has a stack of 8 cards numbered from 1 to 8 she randomly chooses a card and without replacing the first card drawn randomly chooses a second card. what is the probability that Stephanie would pick two cards that are prime numbers

Answer 1

The stack has 8 cards numbered from 1 to 8

She chooses one card and then, without replacing she chooses a second card without replacing the first card chosen.

The possible outcomes are {1, 2, 3, 4, 5, 6, 7, 8}

Out of these numbers, 4 are prime numbers {2, 3, 5, 7}

To determine the probability of chosing one card at random and this card to be a prime number, you have to divide the number of cards that are prime numbers by the total number of cards.

Let "A" represent the event "The first card is a prime number", the probability will be:

`P(A)=(4)/(8)=(1)/(2)`

Now that we took one card from the stack, the total number of cards is reduced from 8 to 7

And because the number was a prime number, to amount of cards with prime numbers is also reduced, from 4 to 3

Let "B" represent the event "the second card is a prime number, given that the first card drawn was a prime number"

You can calculate the probability as:

`P(B|A)=(3)/(7)`

Finally what we need to calculate is the probability that "The first card is a prime number" and "the second card is a prime number, considering that the first card was a prime number"

This is an intersection and you can calculate it as the product between both events:

`P(A)\cdot P(B|A)=(1)/(2)\cdot(3)/(7)=(3)/(14)`

The probability that she would pick two cards that are prime numbers is 3/14 ≅ 0.21