QuestionFind the probability of drawing an even or prime card from a 52-card deck. Express your answer as a simplified fraction.

Studdy Solution

STEP 1

Assumptions1. A standard deck of52 playing cards is used. . The deck contains4 suits hearts, diamonds, clubs, and spades.
3. Each suit has13 cards Ace,,3,4,5,6,7,8,9,10, Jack, Queen, and King.
4. The even cards are,4,6,8, and10.
5. The prime numbers are,3,5, and7.
6. Ace, Jack, Queen, and King are neither even nor prime.
7. We are looking for the probability of drawing a card that is either even or prime.

STEP 2

First, we need to find the total number of even cards in the deck. Since there are4 suits and each suit has5 even cards (2,4,6,8, and10), we multiply the number of suits by the number of even cards per suit.
$Even\, cards = Number\, of\, suits \\times Number\, of\, even\, cards\, per\, suit$

STEP 3

Now, plug in the given values for the number of suits and the number of even cards per suit to calculate the total number of even cards.
$Even\, cards = \\times5$

STEP 4

Calculate the total number of even cards.
$Even\, cards =4 \\times =20$

STEP 5

Next, we need to find the total number of prime cards in the deck. Since there are4 suits and each suit has4 prime cards (2,3,5, and7), we multiply the number of suits by the number of prime cards per suit.
$Prime\, cards = Number\, of\, suits \\times Number\, of\, prime\, cards\, per\, suit$

STEP 6

Now, plug in the given values for the number of suits and the number of prime cards per suit to calculate the total number of prime cards.
$Prime\, cards =4 \\times4$

STEP 7

Calculate the total number of prime cards.
$Prime\, cards =4 \\times4 =16$

STEP 8

Now, we need to subtract the cards that are both even and prime because we have counted them twice. The only card that is both even and prime is2, and there are4 of them in the deck (one in each suit).
$Double\, counted =4$

STEP 9

Now, we can find the total number of cards that are either even or prime by adding the total number of even cards and the total number of prime cards, then subtracting the double counted cards.
$Even\, or\, prime = Even\, cards + Prime\, cards - Double\, counted$

STEP 10

Plug in the values for the even cards, prime cards, and double counted cards to calculate the total number of cards that are either even or prime.
$Even\, or\, prime =20 +16 -4$

STEP 11

Calculate the total number of cards that are either even or prime.
$Even\, or\, prime =20 +16 -4 =32$

STEP 12

Now that we have the total number of cards that are either even or prime, we can find the probability of drawing one of these cards by dividing the total number of even or prime cards by the total number of cards in the deck.
$Probability = Even\, or\, prime / Total\, number\, of\, cards$

STEP 13

Plug in the values for the even or prime cards and the total number of cards to calculate the probability.
$Probability =32 /52$

STEP 14

implify the fraction to get the probability.
$Probability =32 /52 =8 /13$ The probability of drawing a card that is either even or prime from a standard deck of52 playing cards is8/13.

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