The number of immigrants living in a certain country with a large population makes up 35% of the population. In a random sample poll of 40 people what is:
a)the expected number of non-immigrants will be polled?
b)the probability that no immigrants will be polled?
c)the probability that at least 3 immigrants will be polled?
Thanks.
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To answer these questions, we need to understand the concept of probability and apply it to the given situation.
a) The expected number of non-immigrants to be polled can be calculated by multiplying the total population by the proportion of non-immigrants in the population. In this case, 35% of the population consists of immigrants, so the remaining 65% would be non-immigrants.
Expected number of non-immigrants = Total population * Proportion of non-immigrants
= 40 * 0.65
= 26
Therefore, the expected number of non-immigrants to be polled is 26.
b) To calculate the probability that no immigrants will be polled, we need to find the complement of the probability that at least one immigrant will be polled.
Probability of no immigrants = 1 - Probability of at least one immigrant
Given that the proportion of immigrants in the population is 35%, the probability that at least one immigrant will be polled can be calculated using the concept of complement:
Probability of at least one immigrant = 1 - Probability of none of the immigrants being polled
Assuming the sample is selected randomly, the probability of not selecting an immigrant from the sample can be determined using the principle of multiplication since each selection is independent:
Probability of not selecting an immigrant = (1 - Probability of selecting an immigrant) per selection
Probability of not selecting an immigrant on any single trial = 1 - 0.35 = 0.65
Since there are 40 trials in this case, the probability that no immigrants will be polled can be calculated by multiplying the probability of not selecting an immigrant on each trial:
Probability of no immigrants = (0.65) ^ 40
≈ 0.00082 (approximately)
Therefore, the probability that no immigrants will be polled in the random sample poll is approximately 0.00082.
c) To calculate the probability that at least 3 immigrants will be polled, we can use the concept of cumulative probability.
Probability of at least 3 immigrants = 1 - Probability of fewer than 3 immigrants
The probability of fewer than 3 immigrants being polled can be calculated using the binomial probability formula:
Probability of fewer than 3 immigrants = P(X = 0) + P(X = 1) + P(X = 2)
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
where n is the total number of trials (40 in this case), k is the number of successful trials (in this case, the number of immigrants), and p is the probability of success (proportion of immigrants in the population).
Using this formula, we can calculate the probability of fewer than 3 immigrants:
P(X = 0) = (40 C 0) * 0.35^0 * (1 - 0.35)^(40 - 0)
P(X = 1) = (40 C 1) * 0.35^1 * (1 - 0.35)^(40 - 1)
P(X = 2) = (40 C 2) * 0.35^2 * (1 - 0.35)^(40 - 2)
By summing these probabilities, we can find the probability of fewer than 3 immigrants being polled. Subtracting this value from 1 will give us the probability of at least 3 immigrants being polled.
Note: (n C k) represents the binomial coefficient, which is calculated as n! / (k! * (n - k)!), where "!" denotes factorial.
I hope this helps! Let me know if you have any further questions.