• A medical center has 8 ambulances. Given the ambulance’s current condition, regular maintenance, and restocking of medical supplies, the probability of an ambulance being operational is 0.96. Find the probability that at least 6 of the 8 ambulances are operational. We will solve this problem later.

• A probability experiment is a binomial
experiment if both of the following conditions are met:
• The experiment consists of n trials whose outcomes are either successes
(the outcome is the event in question)
or failures (the outcome is not event in question.)

• In a binomial experiment, the trials are identical and independent with a constant probability of success, known as p, and a constant probability of failure, known as 1 – p. In other words, since there is a constant probability of success, this means that the probability of success will be the same for each and every trial of the experiment.

• In a binomial experiment consisting of
n trials, the probability, P, of r successes (where 0 ≤ r ≤ n , p is the probability of success, and 1-p is the probability of failure) is given by the following formula:
P = n
C r p r (1 – p) n-r

• According to a survey taken by USA
TODAY, about 37% of adults believe that
UFOs really exist. Suppose you randomly survey 6 adults. What is the probability that exactly 2 of them believe that
UFOs really exists?

• Let p = 0.37 be the probability that a randomly selected adult believes that UFOs really exist.
By survey 6 adults, you are conducting n = 6 independent trials. The probability of getting exactly r = 2 successes is:
• P( r=2) =
6
C
2
(0.37) 2 (1- 0.37) 6-2
6!
2 (0.63) 4
≈ 0.323, or 32%

Constructing a Binomial Distribution
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 1 2 3 4 5 6
Number of Believers in UFOs

• The previous slide showed the binomial distribution in a histogram.
• Here are the solution values for the individual r values:
• P(r = 0) =
6
C
0
(0.37) 0 (0.63) 6 ≈ 0.063
• P(r = 1) =
6
C
1
(0.37) 1 (0.63) 5 ≈ 0.220
• P(r = 2) =
6
C
2
(0.37) 2 (0.63) 4 ≈ 0.323
• P(r = 3) =
6
C
3
(0.37) 3 (0.63) 3 ≈ 0.253
• P(r = 4) =
6
C
4
(0.37) 4 (0.63) 2 ≈ 0.112
• P(r = 5) =
6
C
5
(0.37) 5 (0.63) 1 ≈ 0.026
• P(r = 6) =
6
C
6
(0.37) 6 (0.63) 0 ≈ 0.003
• The probability of getting at most r = 2 successes is:
• P(r ≤ 2) = P(2) + P(1) + P(0) ≈ .323 + .220 + .063 ≈ 0.606
• The probability that at most 2 of the people believed that UFOs really exist is about 61%

• Problem: Find the probability that at least 6 of 8 ambulances are operational. Round to the nearest tenth of a percent.
• At least 6 ambulances are operational when exactly
6,7,or 8 ambulances are operational.
• Find P(exactly 6) + P(exactly 7)+ P(exactly 8)
• Use n = 8, p = 0.96, and 1- p = 1 – 0.96
• Use the Binomial Probability formula that we used in the
UFO problem. It is on slide #4.

• For a science project you are incubating 12 chicken eggs. The probability that a chick is female is 0.5.
• Start by calculating each binomial probability using this formula:
• P (r) = n
C r
(0.5) r (0.5) n-r = n
C r
(0.5) n
• Then draw a histogram of the binomial distribution based on the probability that exactly r of the chicks are female.
• Finally, find the most likely number of female chicks.
• What do you notice about the distribution of bars on your histogram? How does it compare with the UFO histogram?