Question

The distribution of the binomial random variable (X) has the following parameters p = 0.3 and n = 9. Determine P(X greater or equal to 4)

Answer 1

To find P(X greater or equal to 4), we need to calculate the probability of getting 4, 5, 6, 7, 8, or 9 successes in 9 trials.

Using the formula for the probability mass function of a binomial distribution:

P(X = k) = nCk * p^k * (1-p)^(n-k)

where nCk is the combination formula "n choose k" and is equal to n! / (k!(n-k)!).

For k = 4:
P(X = 4) = 9C4 * 0.3^4 * (1-0.3)^(9-4) = 126 * 0.01287 * 0.3269 ≈ 0.0513

For k = 5:
P(X = 5) = 9C5 * 0.3^5 * (1-0.3)^(9-5) = 126 * 0.00243 * 0.4131 ≈ 0.0126

For k = 6:
P(X = 6) = 9C6 * 0.3^6 * (1-0.3)^(9-6) = 84 * 0.000729 * 0.729 ≈ 0.0435

For k = 7:
P(X = 7) = 9C7 * 0.3^7 * (1-0.3)^(9-7) = 36 * 0.0002187 * 0.81 ≈ 0.0053

For k = 8:
P(X = 8) = 9C8 * 0.3^8 * (1-0.3)^(9-8) = 9 * 0.0000656 * 0.9 ≈ 0.0005

For k = 9:
P(X = 9) = 9C9 * 0.3^9 * (1-0.3)^(9-9) = 1 * 0.0000196 * 1 ≈ 0.00002

Now, adding up the probabilities for k = 4, 5, 6, 7, 8, and 9:

P(X greater or equal to 4) = 0.0513 + 0.0126 + 0.0435 + 0.0053 + 0.0005 + 0.00002
≈ 0.1132

Therefore, P(X greater or equal to 4) is approximately 0.1132.