Binomial Distributions Worksheet 1

You may wish to use the interactive binomial probabilities page. This page duplicates and expands on the binomial tables. (Opens in a separate browser window.)

P1. For a binomial variable X, with n = 5 and p = 0.40.

  1. Use the cumulative probabilities (tabled) to find f(x) = P[ X = x ] for each possible value of X.

    x  
    f(x)  
  2. Draw a histogram.
  3. Compute the mean and standard deviation of X. Locate the mean on the horizontal axis your histogram.

P2. For a binomial variable X, with n = 14 and p = 0.80, find

  1. P[ X <= 8 ]. (<= is the "less than or equal to" sign)
  2. P[ X = 8 ].
  3. P[ X >= 12 ].
  4. The mean and standard deviation of X.
  5. P[ 9 <= X <= 13 ].

P3. For the binomial distribution with n = 10 and p = 0.3, find the probability of

  1. Five or more successes.
  2. At most two successes.
  3. At least one success.
  4. At least 50% successes.

P4. Find the probability of

  1. 12 successes in 20 trials when the probability of a success is 0.7.
  2. 8 failures in 20 trials when the probability of a failure is 0.3.
  3. Explain why you get identical answers in parts (a) and (b).

P5. The population of a large city is 60% African-American. A jury of 12 is selected at random from the citizens of the city.

  1. What is the probability that at most 3 African-Americans are selected?
  2. What is the probability that at most 2 African-Americans are selected?
  3. What is the expected number of African-Americans on the jury? What is the standard deviation?

P6. Suppose, for the purpose of argument, that 20% of the otters in a large aquatic community are infected with a parasite. A biologist samples 16 otters at random and records, for each, whether or not the animal is infected.

  1. Find the mean number of infected otters in the sample.
  2. What is the probability exactly 20% of the sampled otters are infected?
  3. What is the probability more than 50% of the sampled otters are infected? To get a better feel for this probability, compute its reciprocal.
  4. Suppose more than 50% of the sampled otters are infected. How might the biologist respond to such information? Use the probability of part (d) to support your argument.

P7. When an assembly machine is properly calibrated approximately 5% of the finished products are defective. Periodically the quality control engineer randomly selects 20 products at random, and tests them. He will halt the production line to check (and reconfigure, if necessary) the calibration if any of the tested products are defective. Assuming that the machine is properly calibrated, what is the probability that production must be halted?