Respuesta :
Answer:
(a) The probability that all three tosses land on heads given that the 1st toss lands on Heads is [tex]\frac{1}{4}[/tex].
(b) The probability that all three tosses land on heads given that at least one toss lands on heads is [tex]\frac{1}{7}[/tex].
Step-by-step explanation:
The sample space of tossing 3 coins is:
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} = 8 outcomes.
(a)
The sample space such that the 1st toss lands on Heads is:
S₁ = {HHH, HHT, HTH, HTT} = 4 outcomes.
The probability that all three tosses land on heads given that the 1st toss lands on Heads is:
P (3 Heads | 1st toss is Heads) = [tex]\frac{P(3\ Heads\ \cap \ 1st\ toss\ is\ Heads)}{P(1st\ toss\ is\ Heads)}[/tex]
[tex]=\frac{\frac{1}{4} }{\frac{4}{8} } \\=\frac{1}{8}\times \frac{8}{4}\\ =\frac{1}{4}[/tex]
Thus, the probability that all three tosses land on heads given that the 1st toss lands on Heads is [tex]\frac{1}{4}[/tex].
(b)
The sample space such that at least one toss lands on heads is:
S₂ = {HHH, HHT, HTH, THH, HTT, THT, TTH} = 7 outcomes
The probability that all three tosses land on heads given that at least one toss lands on heads is,
P (3 Heads | At least 1 Heads) [tex]=\frac{P(3\ Heads\ \cap\ At\ least\ 1\ Heads)}{P( At\ least\ 1\ Heads)}[/tex]
[tex]=\frac{\frac{1}{7} }{\frac{7}{8} } \\=\frac{1}{8}\times \frac{8}{7}\\ =\frac{1}{7}[/tex]
Thus, the probability that all three tosses land on heads given that at least one toss lands on heads is [tex]\frac{1}{7}[/tex].
