Lesson Worksheet: Binomial Distribution Mathematics

In this worksheet, we will practice identifying binomial experiments and solving probability problems of binomial random variables.

Q1:

Decide if the following experiment is a binomial experiment: drawing cards from a deck of 52 cards to see how many times the number 5 appears.

If it is, state the probability of a success as a fraction.

  • AA binomial experiment, with 𝑝=113
  • BNot a binomial experiment

Q2:

Follow the steps to construct a binomial experiment to find the experimental probability of rolling two dice and obtaining two numbers which sum to more than 10.

One trial of the experiment would be rolling two dice and we will conduct 25 trials. How would we define success for each trial?

  • AGetting two numbers which sum to less than 10
  • BGetting two numbers which sum to 10
  • CGetting two numbers which sum to more than 10

State the probability, 𝑝, of a success as a fraction in its simplest form.

  • A13
  • B112
  • C16
  • D12
  • E136

State the probability of a failure.

  • A56
  • B12
  • C1112
  • D23
  • E3536

Describe the random variable 𝑋 in this experiment, which is binomially distributed.

  • A𝑋 is the number of times we get two numbers which sum to 10 in our 25 trials.
  • B𝑋 is the number of times we get two numbers which sum to more than 10 in our 25 trials.
  • C𝑋 is the number of times we get two numbers which sum to less than 10 in our 25 trials.

Q3:

In a binomial experiment, this spinner is spun 10 times and the result is recorded as a success if the top score is achieved.

Let 𝑋 be the number of successes.

Determine 𝑃(𝑋=2) as a percentage to 3 decimal places.

  • A14.599%
  • B25.028%
  • C0.617%
  • D18.771%
  • E28.157%

Determine 𝑃(𝑋=9) as a percentage to 3 decimal places.

  • A0.008%
  • B1.622%
  • C0.003%
  • D0.005%
  • E28.157%

Q4:

Jacob is playing a game with his friend with a fair die. He will win if the die lands on 5. If Jacob plays 6 games, what is the average number of times he wins?

  • A16
  • B5
  • C1
  • D56
  • E6

Q5:

When a coin is flipped 4 times, what is the probability that no tails appear? Approximate to four decimal places, if needed.

Q6:

A teacher asked 10 students about their birth month. What is the probability that 2Β studentsΒ were born in March? Approximate to three decimal places if needed.

Q7:

Which of the following is not true about the binomial random variable with parameters 𝑛, the number of repetition, and 𝑝, the probability of success?

  • AThe variance of the binomial random variable equals 𝑛𝑝(1βˆ’π‘).
  • BThe binomial random variable is the repetition of a Bernoulli trial 𝑛 independent times.
  • CThe probability of a success in each Bernoulli trial is the same.
  • DThe average value of the binomial random variable equals 𝑛𝑝.
  • EThe binomial random variable is a continuous random variable.

Q8:

3 cards are drawn from a standard deck, and the number of aces is counted.

If the 3 cards are drawn one after the other without replacement, is the number of aces considered a binomial random variable?

  • ANo
  • BYes

If the 3 cards are drawn one after the other with replacement, is the number of aces considered a binomial random variable?

  • AYes
  • BNo

If the 3 cards are drawn one after the other with replacement, what is the probability that we get 2 aces? Approximate to four decimal places if needed.

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