Respuesta :
Answer:
The chance that the cards are not all hearts = [tex]\displaystyle\bf\frac{839}{850}[/tex]
Step-by-step explanation:
The chance that the cards are not all hearts means it may contain 0 heart, 1 heart or 2 hearts. Instead of finding each probability, we can just find the probability of "all 3 cards are hearts" by using the probability without replacement, then subtract the result:
Let:
- A is the event of 1st card is heart
- B is the event of 2nd card is heart
- C is the event of 3rd card is heart
For event A:
n(S)₁ = 52 (total cards of a deck is 52)
n(A) = 13 (total cards that are hearts are 13)
[tex]\displaystyle P(A)=\frac{n(A)}{n(S)_1}[/tex]
[tex]\displaystyle =\frac{13}{52}[/tex]
[tex]\displaystyle =\frac{1}{4}[/tex]
For event B:
n(S)₂ = 51 (1 card is taken without replacement)
n(B) = 12 (1 "heart" card is taken without replacement)
[tex]\displaystyle P(B)=\frac{n(B)}{n(S)_2}[/tex]
[tex]\displaystyle =\frac{12}{51}[/tex]
[tex]\displaystyle =\frac{4}{17}[/tex]
For event C:
n(S)₃ = 50 (2 cards are taken without replacement)
n(C) = 11 (2 "heart" cards are taken without replacement)
[tex]\displaystyle P(C)=\frac{n(C)}{n(S)_3}[/tex]
[tex]\displaystyle =\frac{11}{50}[/tex]
The chance that the all 1st, 2nd and 3rd cards are hearts:
[tex]P(A\cap B\cap C)=P(A)\times P(B)\times P(C)[/tex]
[tex]\displaystyle =\frac{1}{4} \times\frac{4}{17} \times\frac{11}{50}[/tex]
[tex]\displaystyle =\frac{11}{850}[/tex]
The chance that the cards are not all hearts:
[tex]P(A\cap B\cap C)'=1-P(A\cap B\cap C)[/tex]
[tex]\displaystyle =1-\frac{11}{850}[/tex]
[tex]\displaystyle=\frac{839}{850}[/tex]
