Q: A survey from Teenagers Research Unlimited found that 30% of teenage consumers receive their spending money from part-time work. If 5 teenagers are selected at random, what is the probability that exactly 3 of them receive their spending money from part-time work?

$$ P(X = 3) = \binom{5}{3} (0.3)^3$ (0.7)^2$ = 10 \times 0.027 \times 0.49 = 0.1323 $$

Q: If the probability that a randomly selected teenager receives their spending money from part-time work is $$ p = 0.3 $$, what is the expected number of teenagers out of 5 who receive their spending money from part-time work?

$$ E[X] = np = 5 \times 0.3 = 1.5 $$

Q: What is the variance of the number of teenagers out of 5 who receive their spending money from part-time work?

$$ Var(X) = np(1-p) = 5 \times 0.3 \times 0.7 = 1.05 $$

Question 1

A survey from Teenagers Research Unlimited found that 30% of teenage consumers receive their spending money from part-time work. If 5 teenagers are selected at random, what is the probability that exactly 3 of them receive their spending money from part-time work?

Accepted Answer

$$ P(X = 3) = \binom{5}{3} (0.3)^3$ (0.7)^2$ = 10 	imes 0.027 	imes 0.49 = 0.1323 $$

Question 2

Given the same scenario, what is the probability that at least 3 of the 5 selected teenagers receive their spending money from part-time work?

Accepted Answer

$$ P(X \geq 3) = P(X=3) + P(X=4) + P(X=5) $$

Question 3

Calculate $$ P(X \geq 3) $$ for the previous problem, where $$ X  is $$the number of teenagers (out of 5) who receive their spending money from part-time work.

Accepted Answer

$$ P(X \geq 3) = 0.1323 + 0.02835 + 0.00243 = 0.16308 $$

Question 4

Which probability distribution is most appropriate for modeling the number of teenagers (out of 5) who receive their spending money from part-time work?

Accepted Answer

Binomial distribution

Question 5

If the probability that a randomly selected teenager receives their spending money from part-time work is $$ p = 0.3 $$, what is the expected number of teenagers out of 5 who receive their spending money from part-time work?

Accepted Answer

$$ E[X] = np = 5 	imes 0.3 = 1.5 $$

Question 6

What is the variance of the number of teenagers out of 5 who receive their spending money from part-time work?

Accepted Answer

$$ Var(X) = np(1-p) = 5 	imes 0.3 	imes 0.7 = 1.05 $$

Question 7

Suppose you want to find the probability that none of the 5 selected teenagers receive their spending money from part-time work. Which formula would you use?

Accepted Answer

$$ P(X=0) = \binom{5}{0} (0.3)^0$ (0.7)^5$ $$

Question 8

If the probability that a teenager receives their spending money from part-time work increases to 0.5, what is the probability that all 5 selected teenagers receive their spending money from part-time work?

Accepted Answer

$$ P(X=5) = (0.5)^5$ = 0.03125 $$

Question 9

A medical researcher wants to know the probability that exactly 3 out of 5 randomly selected patients visited a doctor last month. If the probability that a patient visited a doctor last month is $$ p $$, which formula should be used?

Accepted Answer

$$ P(X=3) = \binom{5}{3} p^3$ (1-p)^2$ $$

Question 10

If $$ X  is a $$binomial random variable representing the number of successes in $$ n $$ independent trials with probability $$ p  of $$success, what is the formula for the probability of observing exactly $$ k $$ successes?

Accepted Answer

$$ P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$ $$

Question 11

A teacher selects 5 students at random and wants to know the probability that at least one of them has part-time work. If the probability that a student has part-time work is $$ p = 0.3 $$, what is the probability that at least one has part-time work?

Accepted Answer

$$ P(X \geq 1) = 1 - (0.7)^5$ = 1 - 0.16807 = 0.83193 $$

Question 12

If the probability that a teenager receives their spending money from part-time work is $$ p = 0.3 $$, what is the probability that at most 2 out of 5 selected teenagers receive their spending money from part-time work?

Accepted Answer

$$ P(X \leq 2) = P(X=0) + P(X=1) + P(X=2) $$

Question 13

Which of the following is NOT an assumption of the binomial distribution?

Accepted Answer

The probability of success changes from trial to trial.

Question 14

A researcher wants to estimate the probability that at least 3 out of 5 randomly selected teenagers have part-time work. Which real-world factor could violate the independence assumption of the binomial model?

Accepted Answer

Teenagers from the same school may influence each other's employment status.

Question 15

If the probability that a teenager receives their spending money from part-time work is $$ p = 0.3 $$, what is the probability that exactly 0 out of 5 selected teenagers receive their spending money from part-time work?

Accepted Answer

$$ P(X=0) = (0.7)^5$ = 0.16807 $$

Question 16

If you want to use the normal approximation to the binomial distribution for $$ n = 5 $$ and $$ p = 0.3 $$, which condition is NOT satisfied?

Accepted Answer

$$ np $$ and $$ n(1-p) $$ are both less than 5.

Probability and the Binomial Distribution: Applications in Survey Data

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