Mathematics, 15.02.2020 01:49 ayoismeisalex

A 3-card poker hand is dealt at random from a standard 52-card deck. What is the total number of possible hands? What is the total number of possible hands if the hand contains exactly one heart?

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Answer from: aaron2113

What is the total number of possible hands?

$nCx = \frac{n!}{x! (n-x)!}$

For this case n = 52 and x =3 and if we replace we got:

$\frac{52!}{3! (52-3)!}= \frac{52!}{3! 49!}= \frac{52*51*50*49!}{3! 49!}= \frac{52*51*50}{3*2*1}= 22100$

So then we have a total of 22100 ways in order to select 3 cards from a total of 52 cards.

What is the total number of possible hands if the hand contains exactly one heart?

For the possible cases we can do this:

$(13C1) *(39C2)= \frac{13!}{1! 12!} \frac{39!}{2! 37!}= 13 \frac{39*38*37!}{2 37!}= 13 *\frac{39*38}{2}= 13*741 =9633$

Since we have 13 hearts and we want to select 1 and the two remain cards are non hearts. So we have 9633 ways in order to have only one heart in a hand of 3, and the probability would be:

$p = \frac{9633}{22110}= 0.436$

Step-by-step explanation:

What is the total number of possible hands?

For this case we assume that w ehave an standard deck of 52 cards and we want to know on how many ways we can select 3 cards from the total of 52, so for this case since the order no matter we can use the formula for combination given by:

$nCx = \frac{n!}{x! (n-x)!}$

For this case n = 52 and x =3 and if we replace we got:

$\frac{52!}{3! (52-3)!}= \frac{52!}{3! 49!}= \frac{52*51*50*49!}{3! 49!}= \frac{52*51*50}{3*2*1}= 22100$

So then we have a total of 22100 ways in order to select 3 cards from a total of 52 cards.

What is the total number of possible hands if the hand contains exactly one heart?

For this case we need to remember that in a tandard deck we have 13 hearts and we want that in the 3 selected card we have just one heart, and we can find this probability using the definition of empirical probability:

$p =\frac{Possible}{Total}$

Where the total cases on this case are 22100

For the possible cases we can do this:

$(13C1) *(39C2)= \frac{13!}{1! 12!} \frac{39!}{2! 37!}= 13 \frac{39*38*37!}{2 37!}= 13 *\frac{39*38}{2}= 13*741 =9633$

Since we have 13 hearts and we want to select 1 and the two remain cards are non hearts. So we have 9633 ways in order to have only one heart in a hand of 3, and the probability would be:

$p = \frac{9633}{22110}= 0.436$

Answer from: Quest

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove.

an inaccurate proof leads to errors, or stating results that are not accurate. which wil mess the probelm up you dont want this

Answer from: Quest

positive

step-by-step explanation:

the quadrant is facing upwards

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