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Given in the question that, Independent flips of a coin that lands on heads with probability $p$are made.

We need to find what is the probability that the first four outcomes are$H,H,H,H$.

As per the given data, According to Independent flips of a coin.

We consider the probability for $H,H,H,H$is $P_4$.

Hence the probability that the first four outcomes are $H,H,H,H$ is$P_4$.

the probability that the first four outcomes are $H,H,H,H$ is $P_4$.

Given in the question that, Independent flips of a coin that lands on heads with probability $p$are made.

We need to find what is the probability that the first four outcomes are$\mathrm{T},\mathrm{H},\mathrm{H},\mathrm{H}$

According to Independent flips of a coin.

We Consider the probability for $T,H,H,H$is $P_3(1鈭�p)$

Hence the probability that the first four outcomes are $T,H,H,H$ is$P_3(1鈭�p)$

the probability that the first four outcomes are$P_3(1鈭�p)$

Given in the question that, Independent flips of a coin that lands on heads with probability $p$are made.

We need to find What is the probability that the pattern $T,H,H,H$occurs before the pattern H, H, H, H?

Lets assume $H,H,H,H$sequence appeared first. Say that $n$is the numbered toss first heads in the first appearance of the $H,H,H,H$sequence.

$H,H,H,H$being before $T,H,H,H$is equivalent to$H_1{H}_2{H}_3{H}_4$The probability of that is$p^4$. since the probability $1$either $H,H,H,H$or $T,H,H,H$occurs, the probability that $T,H,H,H$occur first is $1-p^4$

The probability that the pattern $T,H,H,H$occurs before the pattern $H,H,H,H$is$1-p^4$