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 x be the number of adults who are against using animals for research in a random sample of two adults. Obtain the probability distribution of x.
To obtain the probability distribution of x, we need to consider the possible values that x can take and the corresponding probabilities.
In this case, x represents the number of adults who are against using animals for research in a random sample of two adults.
We have two possible values for x:
1. x = 0: This means that neither of the two adults in the sample is against using animals for research.
2. x = 1: This means that one of the two adults in the sample is against using animals for research.
To find the probability of each of these values, we can use the binomial probability formula:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- n is the total number of trials (in this case, the number of adults in the sample, which is 2).
- x is the number of successful outcomes (in this case, the number of adults against using animals for research).
- p is the probability of a successful outcome (in this case, the proportion of adults against using animals for research, which is 0.35).
- (nCx) represents the number of ways to choose x items from a set of size n, calculated as n! / (x!(n-x)!).
Calculating the probabilities:
For x = 0:
P(x=0) = (2C0) * (0.35)^0 * (1-0.35)^(2-0)
= 1 * 1 * 0.65^2
= 0.4225
For x = 1:
P(x=1) = (2C1) * (0.35)^1 * (1-0.35)^(2-1)
= 2 * 0.35 * 0.65^1
= 0.455
Therefore, the probability distribution of x is:
P(x=0) = 0.4225
P(x=1) = 0.455