W12 Homework B

Due date: Tuesday 4/7, 11:59pm

Summations

01 $★$

03

Counting flip flops

A bag contains 50 marbles, 30 blue and 20 red. A sequence of zeros and ones is created by pulling the marbles out one at a time (without replacement) and writing a 1 if the marble drawn is blue and a zero if it is red.

How many pairs of adjacent digits in the sequence are expected to differ from each other?

Hint: Use a sum of 49 indicators.

Link to original

Solution

Solutions---5270-03

Central Limit Theorem

02 $★$

11

Community college ages

At a community college, the mean age of the students is 22.3 years, and the standard deviation is 4 years. A random sample of 64 students is drawn.

What is the probability that the average age of the students in the random sample is less than 23 years?

Link to original

Solution

Solutions---5280-11

03

03

Normal approximation - Ventilator filters

A mechanical ventilator model uses air filters that last 100 hours on average with a standard deviation of 30 hours.

How many filters should be stocked so that the supply lasts 2,000 hours with probability at least 95%? Use a normal approximation to estimate the answer.

State the reason that the normal approximation is applicable.

Link to original

Solution

Solutions---5280-03

04

12

Winning the lottery

Suppose a lottery game requires that you purchase a $10 game card and advertises a 10% probability of winning a prize.

Use the Central Limit Theorem to approximate the probability of winning at least 20 times when you purchase 100 of these game cards.

Link to original

Solution

Solutions---5280-12

05 $★$

08

Normal approximation - Heads v. tails

Flip a coin 10,000 times. Let $H$ measure the number of heads, and $T$ measure the number of tails. Estimate the probability that $H$ and $T$ are within 100 of each other.

Hint: Write an inequality for the condition, then sub a formula for $T$ in terms of $H$.

Link to original

Solution

Solutions---5280-08

06

05

Indicator method, exchangeability, summation rules

A class has 40 students: 24 women and 16 men. Each period the teacher selects a random student to present an exercise on the board from among those who have not presented already.

Let $X$ count the number of times a man was chosen after 15 class periods.

(a) Find $E[X]$.

(b) Find $\mathrm{Var}[X]$.

Hint: Is $X_j$ independent of $X_i$? Do you know $E[X_j]$ anyway?

Link to original

Solution

Solutions---5280-05