Q.4.50 - A First Course in Probability

Question

Suppose that a biased coin that lands on heads with probability $p$ is flipped $10$ times. Given that a total of $6$ heads results, find the conditional probability that the first $3$ outcomes are

(a) $h, t, t$ (meaning that the first flip results in heads, the second is tails, and the third in tails);

(b) $t, h, t.$

Step-by-Step Solution

Answer

The conditional probability that the first $3$ outcomes are $h, t, t$ is $\frac{1}{10}.$

The conditional probability that the first $3$ outcomes are $t, h, t$ is $\frac{1}{10}.$

Step 1: Given Information (Part-a)

From the information, observe that a biased coin that lands on heads with probability $p$ is flipped 10 times.

Number of times a coin is tossed, $n=10$

Number heads, $h=6$

A biased coin that lands on heads with probability is $P.$

Let $X$ denote getting outcome.

Here, the random variable $X$ follows a binomial distribution with probability of success $p$ and the number of trials $10$.

The probability mass function of binomial distribution can be defined as,

$P(X=h)=\left(10C_h\right)p^h(1-p{)}^{10-h}$

Step 2: Solution of the Problem (Part-a)

Calculate the conditional probability that the first $3$ outcomes are $h, t, t.$

That is find $P(h, t, t \backslash 6 h).$

$P(h,t,t\mid 6h)=\frac{P(h,t,t\cap 6h)}{P\left(6h\right)}$

$=\frac{P(h,t,t)P\left(5\text{ heads occur in the last }7\text{ trials }\right)}{P\left(6h\right)}$

$=\frac{p\times (1-p)\times (1-p)\times \left(7C_5\right)p^5(1-p{)}^2}{\left(10C_6\right)p^6(1-p{)}^4}$

We get,

$=\frac{7C_5}{10C_6}$

$=\frac{1}{10}$

Step 3: Final Answer (Part-a)

Therefore, the conditional probability that the first 3 outcomes are $h, t, t$ is$\frac{1}{10}.$

Step 4: Given Information (Part-b)

Observing that a biased coin that lands on heads with probability is flipped times from the information is flipped $10$ times.

Number of times a coin is tossed,$n=10$

Number heads, $h=6$

A biased coin that lands on heads with probability is $p.$

Let $X$denotes getting outcome.

Here, the random variable $X$ follows a binomial distribution with probability of success $p$ and the number of trials $10 .$

The probability mass function of binomial distribution can be defined as,

$P(X=h)=\left(10C_h\right)p^h(1-p{)}^{10-h}$

Step 5: Solution of the problem (Part-b)

Find the conditional probability that the first $3$ outcomes are$t, h, t.$

That is find $P(t,h,t\mid 6h)$

$P(t,h,t\mid 6h)=\frac{P(t,h,t\cap 6h)}{P\left(6h\right)}$

$=\frac{P(t,h,t)\times P\left(5\text{ heads occur in the last }7\text{ trials }\right)}{P\left(6h\right)}$

$=\frac{(1-p)\times p\times (1-p)\times \left(7C_5\right)p^3(1-p{)}^2}{\left(10C_6\right)p^6(1-p{)}^4}$

We get,

$=\frac{7C_5}{10C_6}$

$=\frac{1}{10}$

Step 6: Final Answer (Part-b)

Therefore, the conditional probability that the first$3$outcomes are $t, h, t$ is $\frac{1}{10}.$