Question

If x is a binomial random variable, compute P(x) for each of the following cases: (a) P(x≤3),n=4,p=0.3 P(x)= (b) P(x>1),n=6,p=0.1 P(x)= (c) P(x<1),n=9,p=0.8 P(x)= (d) P(x≥2),n=8,p=0.1 P(x)=

Answer 1

To compute the probability in each case, we can use the binomial distribution formula. The binomial distribution gives the probability of obtaining a certain number of successes in a fixed number of independent trials.

Step-by-step explanation:

To compute the probability in each case, we can use the binomial distribution formula. The binomial distribution gives the probability of obtaining a certain number of successes in a fixed number of independent trials. The formula for P(x) in the binomial distribution is P(x) = nCx * (p^x) * (q^(n-x)), where nCx represents the number of ways to choose x successes out of n trials, p is the probability of success, and q is the probability of failure.

1. (a) P(x≤3), n=4, p=0.3: P(x≤3) = P(x=0) + P(x=1) + P(x=2) + P(x=3)
   P(x=0) = 4C0 * (0.3^0) * (0.7^4) = 1 * 1 * 0.2401 = 0.2401
   P(x=1) = 4C1 * (0.3^1) * (0.7^3) = 4 * 0.3 * 0.343 = 0.4116
   P(x=2) = 4C2 * (0.3^2) * (0.7^2) = 6 * 0.09 * 0.49 = 0.2646
   P(x=3) = 4C3 * (0.3^3) * (0.7^1) = 4 * 0.027 * 0.7 = 0.0948
   P(x≤3) = 0.2401 + 0.4116 + 0.2646 + 0.0948 = 1.0111
2. (b) P(x>1), n=6, p=0.1: P(x>1) = 1 - P(x≤1)
   P(x=0) = 6C0 * (0.1^0) * (0.9^6) = 1 * 1 * 0.5314 = 0.5314
   P(x=1) = 6C1 * (0.1^1) * (0.9^5) = 6 * 0.1 * 0.5905 = 0.3543
   P(x≤1) = 0.5314 + 0.3543 = 0.8857
   P(x>1) = 1 - 0.8857 = 0.1143
3. (c) P(x<1), n=9, p=0.8: P(x<1) = P(x=0)
   P(x=0) = 9C0 * (0.8^0) * (0.2^9) = 1 * 1 * 0.001 = 0.001
4. (d) P(x≥2), n=8, p=0.1: P(x≥2) = 1 - P(x<2)
   P(x=0) = 8C0 * (0.1^0) * (0.9^8) = 1 * 1 * 0.4305 = 0.4305
   P(x=1) = 8C1 * (0.1^1) * (0.9^7) = 8 * 0.1 * 0.4783 = 0.3826
   P(x<2) = 0.4305 + 0.3826 = 0.8131
   P(x≥2) = 1 - 0.8131 = 0.1869

Answer 2

(a) P(x ≤ 3) = 0.8257

(b) P(x > 1) = 0.9650

(c) P(x < 1) = 0.0064

(d) P(x ≥ 2) = 0.7179

Step-by-step explanation:

For each case, we need to use the binomial probability formula:

P(x) = nCx * p^x * (1-p)^(n-x)

where:

n is the number of trials

p is the probability of success in each trial

x is the number of successes we are interested in

nCx is the binomial coefficient, which can be calculated using a calculator or statistical software

Calculations:

(a) P(x ≤ 3) = 4C3 * 0.3^3 * 0.7 + 4C2 * 0.3^2 * 0.7^2 + 4C1 * 0.3^1 * 0.7^3 + 4C0 * 0.3^0 * 0.7^4 ≈ 0.8257

(b) P(x > 1) = 1 - P(x ≤ 1) = 1 - (6C0 * 0.1^0 * 0.9^6 + 6C1 * 0.1^1 * 0.9^5) ≈ 0.9650

(c) P(x < 1) = P(x = 0) = 9C0 * 0.8^0 * 0.2^9 ≈ 0.0064

(d) P(x ≥ 2) = 1 - P(x < 2) = 1 - (8C0 * 0.1^0 * 0.9^8 + 8C1 * 0.1^1 * 0.9^7) ≈ 0.7179

These calculations provide the probabilities for each case based on the given parameters.