The distribution of the binomial random variable (X) has the following parameters p = 0.3 and n = 9. Determine P(X greater or =4)
To determine P(X greater or equal to 4), we need to find the probability of getting 4, 5, 6, 7, 8, or 9 successes in 9 trials.
Using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where (n choose k) = n! / (k! * (n-k)!)
P(X=4) = (9 choose 4) * 0.3^4 * 0.7^5 = 126 * 0.3^4 * 0.7^5 = 0.0583
P(X=5) = (9 choose 5) * 0.3^5 * 0.7^4 = 126 * 0.3^5 * 0.7^4 = 0.1367
P(X=6) = (9 choose 6) * 0.3^6 * 0.7^3 = 84 * 0.3^6 * 0.7^3 = 0.1852
P(X=7) = (9 choose 7) * 0.3^7 * 0.7^2 = 36 * 0.3^7 * 0.7^2 = 0.1323
P(X=8) = (9 choose 8) * 0.3^8 * 0.7^1 = 9 * 0.3^8 * 0.7^1 = 0.0548
P(X=9) = (9 choose 9) * 0.3^9 * 0.7^0 = 1 * 0.3^9 * 1 = 0.009
Now, sum up the probabilities of getting 4, 5, 6, 7, 8, or 9 successes:
P(X greater or equal to 4) = P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) = 0.0583 + 0.1367 + 0.1852 + 0.1323 + 0.0548 + 0.009 = 0.5763
Therefore, the probability of getting 4 or more successes in 9 trials with success probability p=0.3 is 0.5763.