Question

The probability of getting heads from throwing a fair coin is 1/2 The fair coin is tossed 4 times. What is the probability that exactly 3 heads occur? 1/4 The fair coin is tossed 4 times. What is the probability that exactly 3 heads occur given that the first outcome was a head? 3/8 The fair coin is tossed 4 times. What is the probability that exactly 3 heads occur given that the first outcome was a tail? 1/8 The fair coin is tossed 4 times. What is the probability that exactly 3 heads occur given that the first two outcomes were heads? 1/2 The fair coin is tossed 4 times. What is the probability that exactly 3 heads occur given that there is at least 1 head? 4/15 A coin is rigged so that the probability of heads is 2/7 The rigged coin is tossed 4 times. What is the probability that exactly 3 heads occur? The rigged coin is tossed 4 times. What is the probability that exactly 3 heads occur given that the first outcome was a head? The rigged coin is tossed 4 times. What is the probability that exactly 3 heads occur given that the first outcome was a tail? The rigged coin is tossed 4 times. What is the probability that exactly 3 heads occur given that the first two outcomes were heads? The rigged coin is tossed 4 times. What is the probability that exactly 3 heads occur given that there is at least 1 head?

Answer 1

for rigged coin:

1)P(exactly 3 heads) =P(X=3) =$\binom{4}{3}(2/7)^{3}(5/7)^{1}$ =0.0666

2)P(2 head in next 3 flips)=$\binom{3}{2}(2/7)^{2}(5/7)^{1}$ =0.1749

3)P(3 head in 3 flips) =(2/7)³ =0.0233

4)P(1 head in next 2 flips) =0.4082

5) P(X=3|X>=1)=0.0901