Worksheet: Binomial Distribution

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.

  • A 1 3
  • B 1 1 2
  • C 1 6
  • D 1 2
  • E 1 3 6

State the probability of a failure.

  • A 5 6
  • B 1 2
  • C 1 1 1 2
  • D 2 3
  • E 3 5 3 6

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

  • A 𝑋 is the number of times we get two numbers which sum to 10 in our 25 trials.
  • B 𝑋 is the number of times we get two numbers which sum to more than 10 in our 25 trials.
  • C 𝑋 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.

  • A 1 4 . 5 9 9 %
  • B 2 5 . 0 2 8 %
  • C 0 . 6 1 7 %
  • D 1 8 . 7 7 1 %
  • E 2 8 . 1 5 7 %

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

  • A 0 . 0 0 8 %
  • B 1 . 6 2 2 %
  • C 0 . 0 0 3 %
  • D 0 . 0 0 5 %
  • E 2 8 . 1 5 7 %

Q4:

What is a Bernoulli trial?

  • AAn experiment whose outcome is random and can be either of two possible outcomes.
  • BAn experiment whose outcome is random integers.
  • CAn experiment whose outcome is random and can be either of three possible outcomes.
  • DAn experiment whose outcome is random continuous numbers between 0 and 1.

Q5:

If the probability of success of a Bernoulli trial is 𝑝, what is the probability that the 𝑘th trial (out of 𝑘 trials) is the first success?

  • A 𝑃 ( 𝑋 = 𝑘 ) = 𝑘 ! 𝑝 ( 1 𝑝 )
  • B 𝑃 ( 𝑋 = 𝑘 ) = 𝑝 ( 1 𝑝 )
  • C 𝑃 ( 𝑋 = 𝑘 ) = ( 1 𝑝 )
  • D 𝑃 ( 𝑋 = 𝑘 ) = 𝑝 ( 1 𝑝 )

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