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 adults let XP the number of adults who are against using animals for research in a random sample of two adults obtain a probability distribution of X

To obtain the probability distribution of X, which represents the number of adults against using animals for research in a random sample of two adults, we can use the concept of binomial distribution.

In this case, we have a binomial experiment because each adult can be classified as either being against using animals for research or not. The probability of success (p) is given as 35%, which can be written as 0.35.

The formula for the probability mass function (PMF) of the binomial distribution is:

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

Where:
- P(X = k) represents the probability of having k successes
- (n choose k) is the binomial coefficient, which can be calculated as n! / (k!(n-k)!)
- p is the probability of success (35% or 0.35 in our case)
- k is the number of successes we want to calculate the probability for
- n is the number of trials or sample size (2 in this case)

Using this formula, we can calculate the probabilities for each possible value of X in the sample of two adults.

For X = 0 (both adults are against using animals for research):
P(X = 0) = (2 choose 0) * 0.35^0 * (1-0.35)^(2-0)
= (1) * 1 * 0.65^2
= 0.65^2
= 0.4225

For X = 1 (one adult is against using animals for research):
P(X = 1) = (2 choose 1) * 0.35^1 * (1-0.35)^(2-1)
= (2) * 0.35 * 0.65^1
= 0.455

For X = 2 (both adults are against using animals for research):
P(X = 2) = (2 choose 2) * 0.35^2 * (1-0.35)^(2-2)
= (1) * 0.35^2 * 1^0
= 0.1225

Therefore, the probability distribution of X for the random sample of two adults is as follows:
P(X = 0) = 0.4225
P(X = 1) = 0.455
P(X = 2) = 0.1225