Respuesta :
Answer:
probability = [tex]\displaystyle\bf\frac{13}{204}[/tex]
Step-by-step explanation:
We can find the probability of choosing a heart then a spade without replacement by using the probability formula:
[tex]\boxed{P(A)=\frac{n(A)}{n(S)} }[/tex]
where:
- [tex]P(A)[/tex] = probability of event A
- [tex]n(A)[/tex] = total outcomes of event A
- [tex]n(S)[/tex] = total outcomes of all possibilities
Let:
- A = drawing a heart at the 1st draw
- B|A = drawing a spade after the event A
For the 1st draw:
- total number of hearts [tex](n(A))[/tex] = 13
- total number of cards [tex](n(S)_A)[/tex] = 52
[tex]\displaystyle P(A)=\frac{n(A)}{n(S)_A}[/tex]
[tex]\displaystyle=\frac{13}{52}[/tex]
[tex]\displaystyle=\frac{1}{4}[/tex]
For the 2nd draw:
- total number of spade [tex](n(B|A))[/tex] = 13
- total number of cards [tex](n(S)_B)[/tex] = 52 - 1 = 51 (1 card is taken without replacement at the 1st draw)
[tex]\displaystyle P(B|A)=\frac{n(B|A)}{n(S)_B}[/tex]
[tex]\displaystyle=\frac{13}{51}[/tex]
[tex]\displaystyle=\frac{1}{4}[/tex]
Hence, for the event "drawing a heart at the 1st draw" AND "drawing a spade after the event A":
[tex]P(A\cup B)=P(A)\times P(B|A)[/tex]
[tex]\displaystyle=\frac{1}{4} \times\frac{13}{51}[/tex]
[tex]\displaystyle=\bf\frac{13}{204}[/tex]
