4.3 Binomial Distributions
Chapter
Chapter 4
Section
4.3
Solutions 23 Videos

Which expression best represents the probability of three successes in seven independent trials in a binomial distribution?

0.46mins
Q1

What is the expectation for a binomial distribution with p = 0.5 and n = 8.?

A. 0.4

B. 4

C. 16

D. 0.0625

0.15mins
Q2

Which of the following is an example of a binomial distribution?

A. probabilities of the number of queens in a five-card hand

B. probability of each sum when two dice are rolled

C. probability of each lane for a 100 m race

D. probabilities of the number of times a 5 occur when spinning a spinner six times

1.08mins
Q3

A tetrahedral die has four triangular faces. Three faces are labelled l and the fourth is labelled 2. The die is rolled four times.

a) Draw a tree diagram to illustrate the possible outcomes.

b) Use the tree diagram to assign probabilities for this distribution.

c) Verify these probabilities using the binomial distribution formula.

d) Substitute values for p and q and expand (p + q)^4, but do not simplify. How does the expansion relate to the above results?

6.28mins
Q4

Prepare a probability table and a graph for a binomial distribution with

n = 6 and p = 0.3

2.49mins
Q5a

Prepare a probability table and a graph for a binomial distribution with

n = 8 and p = \frac{1}{9}

2.10mins
Q5b

What is the expected number of times a 6 appears when rolling a die 2000 times?

0.36mins
Q6

In a family of five children, what is the probability that there are exactly

a) two girls?

b) three boys?

1.13mins
Q7

Six people are asked to choose a number between 1 and 20. What is the probability that

a) two people choose the number 9?

b) at least two people choose the number 9?

2.10mins
Q8

Two dice are rolled repeatedly and their sum is recorded.

a) Show the probability distribution for the number of sums of 7 in five rolls.

b) Graph the distribution with a probability histogram.

c) Verify the formula E(X) = n p

4.49mins
Q9

In archery competitions, Paul hits the bull's-eye 45% of the time.

a) Show the probability distribution for the number of bull’s-eyes in eight attempts.

b) What is the expected number of bull’s-eyes in eight attempts?

c) What does P(8) tell you?

2.30mins
Q10

a) You roll five dice at the same time. What is the probability that you roll two 3s?

b) Expand the binomial \displaystyle (\frac{1}{6} + \frac{5}{6})^5. 

c) Which term in the expansion matches the answer in part a)?

d) How does the binomial probability distribution relate to the binomial theorem?

2.36mins
Q11

A machine makes light bulbs, and 6% do not meet the specifications. An inspector randomly chooses 10 light bulbs for testing.

What is the probability that three bulbs do not meet specifications?

1.10mins
Q12a

A machine makes light bulbs, and 6% do not meet the specifications. An inspector randomly chooses 10 light bulbs for testing.

What is the probability that seven bulbs do not meet specifications?

0.41mins
Q12b

A machine makes light bulbs, and 6% do not meet the specifications. An inspector randomly chooses 10 light bulbs for testing.

What is the probability that between three and seven bulbs do not meet specifications?

2.03mins
Q12c

On a game show, five contestants are each given a box containing 10 car keys. one of which fits their assigned new car.

Each contestant is allowed to choose one key and try to start their car. If no car starts. or only one car starts, nobody wins their car.

If two or more cars start, then those contestants win their car. Do the results of the game favour the contestants or the game show? Justify mathematically.

2.10mins
Q13

Jamaal is successful on basketball free throws 80% of the time.

a) How likely is he to be successful on eight of 10 free-throw attempts?

b) How likely is he to be successful on at least eight of 10 free-throw attempts?

1.29mins
Q14

Jean forgot to study for an eight-question multiple choice quiz. Each question contains four possible answers. Jean will guess the answer to each question.

a) Whatistheprobabilitythatshewillget only two questions correct?

b) What is the probability that she will pass?

c) What is the expected number of correct answers on the quiz?

3.38mins
Q15abc

A jar contains 12 red balls and eight green balls. Six balls are removed without replacement. What is the probability that four of the balls are red?

Explain why the binomial distribution is not a suitable model for this problem.

2.04mins
Q16

Opinion polls based on small samples often yield misleading results. In a particular city, 65% of residents are opposed to a new light rail transit system.

a) If a poll were taken, calculate the probabilities of a majority of people approving the transit system with a sample of

• 7 people
• 100 people
• 1000 people

b) Explain any differences in the results.

3.41mins
Q17

A store offers a scratch and wins a discount for each customer who spends over $100. Each card has six spots that give a discount of$10. three spots that give a discount of $25, and one spot that gives a discount of$50. What is the expected cost to the store if it has 200 customers on one particular day?

1.29mins
Q18

A standard die is painted so that opposite faces are green, red, and yellow, respectively. In 10 rolls of this die, how many could be red or green or yellow? This leads to what could be a trinomial distribution.

a) Develop a formula to calculate the probability distribution in which there are three outcomes with individual probabilities of p, q, and r.

b) Use your formula to determine the probability of rolling three reds, two greens, and five yellows in 10 rolls of the die described above.

Derivetheformulaforexpectationofa binomial distribution E(X) = np algebraically.