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

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

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

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?

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

n = 6 and p = 0.3

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

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

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

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

a) two girls?

b) three boys?

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?

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

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?

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?

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?

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?

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?

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.

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?

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?

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.

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.

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?

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.