Question

There are 3 coins which when flipped come up heads, respectively, with probabilities 1/4, 1/2, 3/4. One of these coins is randomly chosen and continually flipped.

1. (a) Find the expected number of flips until the first head.

2. (b) Find the mean number of heads in the first 8 flips.

Answer 1

Answer:

Let us consider the probability of p to get a head on a toss, the normal number of tosses for the principal head is registered as = 1/p

a)

To determine the expected number of flips until the first head

On the off chance that one of the three coins is haphazardly chosen, expected number of flips until the primary head is chosen is registered as:

= (1/3) * (1/p1 + 1/p2+1/p3)

= 1/3 * (1 /(1/4) + 1/(1/2) + 1/(3/4))

= (1/3) * (4 + 2 4/3)

= 1/3 * 22/3

= 22/9

Expected number of flips until the first head = 22/9

b)

To determine the mean number of heads in the first 8 flips

This is fundamentally the same as the past part, the mean number of heads for a coin with probability of head as p in the initial 8 flips is registered as = 8p

Hence if every one of the three coins are chosen with equivalent likelihood, the mean number of heads in the initial 8 flips is registered as follows

= (1/3) *8* (P1 + P2 + P3)

= (1/3)*8*1/4+1/2+3/4)

= 1/3 * 8 * (0.25 + 0.5 + 0.75)

= 1/3 * 8 * 1.5

= 4

The mean number of heads in the first 8 flips is 4