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1. A report from the Secretary of Health and Human Services stated that 70% of single-vehicle traffic fatalities that occur at night on weekends involve an intoxicated driver. If a sample of 15 single-vehicle fatalities that occur on a weekend night is selected, find the probability that between 10 and 15, inclusive, accidents involved drivers who were intoxicated. (7 points)

Sagot :

Using the binomial distribution, it is found that there is a 0.7215 = 72.15% probability that between 10 and 15, inclusive, accidents involved drivers who were intoxicated.

For each fatality, there are only two possible outcomes, either it involved an intoxicated driver, or it did not. The probability of a fatality involving an intoxicated driver is independent of any other fatality, which means that the binomial distribution is used to solve this question.

Binomial probability distribution

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are:

  • x is the number of successes.
  • n is the number of trials.
  • p is the probability of a success on a single trial.

In this problem:

  • 70% of fatalities involve an intoxicated driver, hence [tex]p = 0.7[/tex].
  • A sample of 15 fatalities is taken, hence [tex]n = 15[/tex].

The probability is:

[tex]P(10 \leq X \leq 15) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)[/tex]

Hence

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 10) = C_{15,10}.(0.7)^{10}.(0.3)^{5} = 0.2061[/tex]

[tex]P(X = 11) = C_{15,11}.(0.7)^{11}.(0.3)^{4} = 0.2186[/tex]

[tex]P(X = 12) = C_{15,12}.(0.7)^{12}.(0.3)^{3} = 0.1700[/tex]

[tex]P(X = 13) = C_{15,13}.(0.7)^{13}.(0.3)^{2} = 0.0916[/tex]

[tex]P(X = 14) = C_{15,14}.(0.7)^{14}.(0.3)^{1} = 0.0305[/tex]

[tex]P(X = 15) = C_{15,15}.(0.7)^{15}.(0.3)^{0} = 0.0047[/tex]

Then:

[tex]P(10 \leq X \leq 15) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) = 0.2061 + 0.2186 + 0.1700 + 0.0916 + 0.0305 + 0.0047 = 0.7215[/tex]

0.7215 = 72.15% probability that between 10 and 15, inclusive, accidents involved drivers who were intoxicated.

A similar problem is given at https://brainly.com/question/24863377

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