The random variable X has a Binomial Distribution B (11, p). If P (X =8) = P (X =7), find the value of p.

0.9800

To find the value of p, we need to use the properties of the binomial distribution and the given information.

The binomial distribution is defined by two parameters, n and p. n represents the number of trials, and p represents the probability of success in each trial.

In this case, the random variable X follows a binomial distribution B(11, p), where n = 11 (number of trials) and p is the unknown probability of success.

The probability mass function (PMF) of the binomial distribution is given by:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

Given that P(X = 8) = P(X = 7), we can write the equation:
(n choose 8) * p^8 * (1 - p)^(11 - 8) = (n choose 7) * p^7 * (1 - p)^(11 - 7)

Simplifying the equation:
(11 choose 8) * p^8 * (1 - p)^3 = (11 choose 7) * p^7 * (1 - p)^4

To simplify further, we can cancel out the binomial coefficients (11 choose 8) and (11 choose 7) since they are equal to (11! / (8! * (11 - 8)!)) = (11! / (7! * (11 - 7)!)) = 11.

So, we have:
p^8 * (1 - p)^3 = p^7 * (1 - p)^4

Dividing both sides by p^7 * (1 - p)^3:
p = (1 - p)

Further simplifying:
2p = 1

Finally, solving for p:
p = 1/2

Therefore, the value of p is 1/2.