A 5-card hand is dealt from a perfectly shuffled deck. Define the events: A: the hand is a four of a kind (all four cards of one rank plus a 5th card). B: at least one of the cards in the hand is an ace Are the events A and B independent?

Kyran Hudson

Kyran Hudson

Answered question

2020-12-30

A 5-card hand is dealt from a perfectly shuffled deck. Define the events: A: the hand is a four of a kind (all four cards of one rank plus a 5th card). B: at least one of the cards in the hand is an ace Are the events A and B independent?

Answer & Explanation

Brittany Patton

Brittany Patton

Skilled2020-12-31Added 100 answers

The four of a kind hand has the pattern AAAAB where A and B are from distinct kinds. The number of such hands =13C1×4C4×12C1×4C1=624 Total number of combinations of different hands =52C5=2598960 P(A=Four of a kind)=(Number of favorable outcomes)/(P(A=Four of a kind)=Number of favorable outcomesTotal number of hands)
=6242598960
=0.000240
P(None of the card is Ace)=Hand with no AceTotal number of hands
=48C5(52)C5
=17123042598960=0.6588
P(Atleast one of the card is Ace)=1P(Hands with no Ace)
=10.6588
P(B)=0.3412 The number of hands having four of a kind and Ace =1C1×4C4×1C1×4C1+12C1×4C4×1C1×4C1
=48+48
=96 Total number of combinations of different hands =52C5=2598960. P(AB)=962598960
=0.000037
P(A)P(B)=0.00024×0.3412
=0.000082 Events A and B are not independent because P(AB)P(A)P(B).

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