Q.3.59 - A First Course in Probability

Question

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?

 (c) What is the probability that the pattern T, H, H, H occurs before the pattern H, H, H, H? 

Step-by-Step Solution

Answer

1. $P_4$
   
2. $P_3(1-P)$
   
3. $I-P_4$
   

Step 1: Given Information (Part-a)

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 $H, H, H, H$.

Step 2: Explanation (Part a)

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 $P_4$.

Step 3:Final Answer (Part-a)

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

Step 4: Given Information(Part-b)

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 $\mathrm{T},\mathrm{H},\mathrm{H},\mathrm{H}$

Step 5 : Explanation (Part-b)

According to Independent flips of a coin. 

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

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

Step 6 :Final Answer(Part-b)

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

Step 7 : Given Information (Part-c)

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?  

Step 8: Explanation(Part-c)

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 $1-p^4$

Step 9:Final Answer(Part-c)

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