Respuesta :
Answer:
What is the total number of possible hands?
[tex] nCx = \frac{n!}{x! (n-x)!}[/tex]
For this case n = 52 and x =3 and if we replace we got:
[tex] \frac{52!}{3! (52-3)!}= \frac{52!}{3! 49!}= \frac{52*51*50*49!}{3! 49!}= \frac{52*51*50}{3*2*1}= 22100[/tex]
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:
[tex] (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[/tex]
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:
[tex] p = \frac{9633}{22110}= 0.436[/tex]
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:
[tex] nCx = \frac{n!}{x! (n-x)!}[/tex]
For this case n = 52 and x =3 and if we replace we got:
[tex] \frac{52!}{3! (52-3)!}= \frac{52!}{3! 49!}= \frac{52*51*50*49!}{3! 49!}= \frac{52*51*50}{3*2*1}= 22100[/tex]
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:
[tex] p =\frac{Possible}{Total}[/tex]
Where the total cases on this case are 22100
For the possible cases we can do this:
[tex] (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[/tex]
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:
[tex] p = \frac{9633}{22110}= 0.436[/tex]
