A hand of 5 cards is dealt to each of

Kaspaueru2

Kaspaueru2

Answered question

2021-12-20

A hand of 5 cards is dealt to each of three players from a standard deck of 52 cards.
(a)What is the probability that one of the players receives all four Aces?
(b)What is the probability that at least one player receives no hearts?

Answer & Explanation

kaluitagf

kaluitagf

Beginner2021-12-21Added 38 answers

a. Let X be the number of Ace cards that a player receives
The probability of getting an Ace p=452=0.0769
n=5
XBinomial(5,0.0769)
P(X=4)=5C4(0.0769)4(10.0769)54=0.00016


b. Let X be the number of heart cards that a player receives
The probability of getting a heart is p=1352=0.25
n=5
XBinomial(5,0.25)
The probability that no player receives no heart :
P(X=0)=5C0(0.25)0(10.25)50=0.2373
The probability that at least one player receives no hearts =1No player receives no heart
=10.2373=0.7627

Lynne Trussell

Lynne Trussell

Beginner2021-12-22Added 32 answers

Use inclusion-exclusion. Call the players A, B, and C.
The probability of only one player getting two aces will be 3 times the probability of any one of them getting two aces minus the instances where two of the players get two aces:
P(A)+P(B)+P(C){P(AB)+P(AC)+P(BC)}
So,
3×(4 choose 2)(48 choose 3)52 choose 53×(4 choose 2)(48 choose 3.3, and42)52 choose 5.5, and 42

nick1337

nick1337

Expert2021-12-28Added 777 answers

There 52 cards and 13 cards in each suit (clubs,spades,heart and diamonds).
We have 13 hearts and 39 other cards. We will find the probability of being dealt at least one heart by using the complement rule. Note the following:
P(being dealt at least one heart)=1P(being dealt no hearts)
=1((39),(5))((52),(5))
0.7785
The probability that a player is dealt 5 cards with at least one heart is is approximately a 0.7785.

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