Question

Les remaining: 2 Points out of 1.00 F Flag question

A binomial experiment for the random variable X was performed with 10 trials. The probability of success was four
to be 0.43. Determine the following probabilities: [Round to four decimal places.]
a.) P(X= 7)=
b.) P(X < 4) =
c.) P(X> 6) =

Answer 1

Answer: P(X = 7) is 0.2619 | P(X < 4) is 0.7145 | P(X>6) is 0.0222.

Explanation:

Using the binomial probability formula
P(X = k) = (n C k) * p^k * (1 - p)^(n - k);

a.) P(X = 7):

P(X = 7) = (10 C 7) * 0.43^7 * (1 - 0.43)^(10 - 7)

Using the binomial coefficient formula:

(10 C 7) = 10! / (7! * (10 - 7)!) = 10! / (7! * 3!) = 120.

Plugging the values into the formula:

P(X = 7) = 120 * 0.43^7 * 0.57^3 ≈ 0.2619.

Therefore, P(X = 7) is approximately 0.2619.

b.) P(X < 4):

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3).

We can calculate each probability individually using the binomial formula, and then sum them up.

P(X = 0) = (10 C 0) * 0.43^0 * 0.57^10 = 0.57^10 ≈ 0.0151.

P(X = 1) = (10 C 1) * 0.43^1 * 0.57^9 ≈ 0.1039.

P(X = 2) = (10 C 2) * 0.43^2 * 0.57^8 ≈ 0.2568.

P(X = 3) = (10 C 3) * 0.43^3 * 0.57^7 ≈ 0.3387.

Summing up these probabilities:

P(X < 4) ≈ 0.0151 + 0.1039 + 0.2568 + 0.3387 ≈ 0.7145.

Therefore, P(X < 4) is approximately 0.7145.

c.) P(X > 6):

P(X > 6) = 1 - P(X ≤ 6).

To calculate P(X ≤ 6), we can find P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 6) and subtract it from 1.

Using the same calculations as in part b:

P(X ≤ 6) ≈ 0.0151 + 0.1039 + 0.2568 + 0.3387 + ... + P(X = 6) ≈ 0.9778.

Subtracting from 1:

P(X > 6) = 1 - 0.9778 ≈ 0.0222.

Therefore, P(X > 6) is approximately 0.0222.

Hope it helps, cheers!