Question

In: Statistics and Probability

5 cards are drawn at random from a standard deck. Give your answer as a fraction...

5 cards are drawn at random from a standard deck. Give your answer as a fraction or decimal out to at least 4 significant digits.

Find the probability that all the cards are hearts.

Find the probability that all the cards are face cards.
Note: Face cards are kings, queens, and jacks.

Find the probability that all the cards are even.
(Consider aces to be 1, jacks to be 11, queens to be 12, and kings to be 13)

Solutions

Expert Solution

Probability of an event = Number of favorable outcomes / Total number of possible outcomes

Let n= 5 cards are drawn at a pack of 52 cards. i.e ⁵²C₅

n= ⁵²C₅

1) A= event of all the cards are hearts.

Total heart cards are 13.

m = ¹³C₅

P(A) = m / n

= ¹³C₅ / ⁵²C₅

ⁿC_r = n! / r ! (n-r) !

P(A) = 1287 / 2598960

= 0.0005

2) B= all the cards are face cards.

face cards- 4 kings, 4 queens, 4 jacks = 12

m = ¹²C₅

P(B) = m / n

= ¹²C₅ / ⁵²C₅

= 792 / 2598960

= 0.0003

3) C= all the cards are even.

(2,4,6,8,10,12).. hearts

(2,4,6,8,10,12)... Diamonds

(2, 4,6,8,10,12).... Clubs

(2,4,6,8,10,12)... Spades

Total cards =24

m = ²⁴C₅

P(C) = m / n

= ²⁴C₅ / ⁵²C₅

= 42504 / 2598960

= 0.0164

orchestra answered 1 year ago

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