Question

Consider 10 independent tosses of a biased coin with the probability of Heads at each toss equal to p , where 0 < p < 1 .

1. Let A be the event that there are 6 Heads in the first 8 tosses. Let B be the event that the 9th toss results in Heads. Find P(B|A) and express it in terms of p.

2. Find the probability that there are 3 Heads in the first 4 tosses and 2 Heads in the last 3 tosses. Express your answer in terms of p.

3. Given that there were 4 Heads in the first 7 tosses, find the probability that the 2nd Heads occurred at the 4th toss. Give a numerical answer.

Answer 1

1. We have 8 coins that, of which 6 will turn heads. Out of 8 Coins, we come up with a numerical representation: 8C6, where C=combination.
Answer: 28⋅p6⋅(1−p)^2*p/(28⋅p6⋅(1-p)^2)
2. Answer= 12p^5*(1-p)^2
3. For the following equation: fill in letters a-f to show problem 3.
ap7(1−p)3+bpc(1−p)d+epf(1−p)f.

a=15
b=60
c=6
d=4
e=10
f=5