Independent flips of a coin that lands on heads with probability p are made. what is the probability that the first four outcomes are (a) h, h,h, h? (b) t, h,h, h? 3 (c) what is the probability that the pattern t, h, h, h occurs before the pattern h, h, h, h? hint for part (c): how can the pattern h, h, h, h occur first?

a) p⁴

b) ( 1 - p)p³

c) 1 - p⁴

Step-by-step explanation:

Data provided in the question:

The coin flips are independent

Probability that coin lands on head, P(H) = p

Thus,

Probability that coin lands on tails, P(T) = ( 1 - p )

Now,

a) P( H,H,H,H ) = P(H) × P(H) × P(H) × P(H)

or

⇒ P( H,H,H,H ) = p × p × p × p = p⁴

b) P( T,H,H,H ) = P(T) × P(H) × P(H) × P(H)

or

⇒ P( T,H,H,H ) = ( 1 - p) × p × p × p = ( 1 - p)p³

c) probability that the pattern T, H, H, H occurs before the pattern H, H, H, H

For obtaining four consecutive Heads (H) we need to get 3 consecutive Heads first.

let we have found our first sequence somewhere in our sequence. Now, observing the throw before the first H. So we can have a Tails (T), i.e sequence THHH will appear first, or we can have no throw because the first H was our first throw.

(It can't be a H otherwise your sequence would not be the first one to occur.)

Thus, the only way for to come first is to start with it, and P() = p⁴.

Hence,

P( T, H, H, H occurs before the pattern H, H, H, H) = 1 - p⁴

a) p⁴

b) ( 1 - p)p³

c) 1 - p⁴

Step-by-step explanation: