According to a survey 35% of adults are against using animals for research. Assume that this result holds true for the current population of all adullts. Let X be the number of adults who are against using animals for research in a random sample of 2 adults.

Find the probability distribution of X

To find the probability distribution of X, we need to determine the possible values of X and their respective probabilities.

Since X represents the number of adults who are against using animals for research in a random sample of 2 adults, the possible values of X are 0, 1, and 2.

To calculate the probabilities, we can use the binomial probability formula:

P(X = k) = C(n, k) * p^k * q^(n-k),

where n is the number of trials (sample size), k is the number of successes (adults who are against using animals for research), p is the probability of success, and q is the probability of failure (1 - p).

In this case, n = 2, k can be 0, 1, or 2, and p = 0.35.

P(X = 0) = C(2, 0) * 0.35^0 * 0.65^(2-0)
= 1 * 1 * 0.65^2
= 0.4225

P(X = 1) = C(2, 1) * 0.35^1 * 0.65^(2-1)
= 2 * 0.35^1 * 0.65^1
= 0.455

P(X = 2) = C(2, 2) * 0.35^2 * 0.65^(2-2)
= 1 * 0.35^2 * 1
= 0.0975

Therefore, the probability distribution of X is as follows:

X = 0 with probability 0.4225
X = 1 with probability 0.455
X = 2 with probability 0.0975

To find the probability distribution of X, we need to determine the possible values that X can take and calculate the probability associated with each value.

Since X represents the number of adults who are against using animals for research, it can take on values from 0 to 2 (since we are sampling 2 adults).

Let's calculate the probabilities for each possible value:

When X = 0:
This means that both adults in the sample are not against using animals for research. The probability of this happening is:
P(X = 0) = (0.65) * (0.65) = 0.4225

When X = 1:
This means that one adult in the sample is against using animals for research, while the other is not. Since there are two possible combinations for this scenario (either the first or second adult is against animal research), we need to calculate the probability for each combination and add them together.
P(X = 1) = (0.35) * (0.65) + (0.65) * (0.35) = 0.455

When X = 2:
This means that both adults in the sample are against using animals for research. The probability of this happening is:
P(X = 2) = (0.35) * (0.35) = 0.1225

Therefore, the probability distribution of X is as follows:

X = 0: P(X = 0) = 0.4225
X = 1: P(X = 1) = 0.455
X = 2: P(X = 2) = 0.1225

P(0) (the probability none are against) = (0.65)^2 = 0.4225

P(1) = 2*(0.65)(0.35) = 0.4550
P(2) (both against) = 0.1225

The sum is 1.00

The "expectation value" (the average of many trials) is 0.70