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35% of all customers who enter a restaurant will order dessert. Suppose that 10 customers enter the restaurant and these customers make independent decisions.
Required:
a) Calculate the probability that exactly five customers order dessert.



Answer :

Answer:

the probability that exactly five customers order dessert ≈ 0.1536

Step-by-step explanation:

We can find the probability that exactly five customers order dessert by using the Binomial Distribution.

Let's check if this question meets the Binomial Distribution's conditions:

  1. fixed number of trials → 10 trials
  2. there is only 2 outcomes → either order dessert or not order dessert
  3. the probability for each trial is constant → 35%
  4. the outcome of each trial is independent

The formula for Binomial Distribution:

[tex]\boxed{P(X=x)=\left(\begin{array}{c}n\\x\end{array}\right)p^xq^{n-x} }[/tex]

where:

  • [tex]P=\text{Probability}[/tex]
  • [tex]x=\text{number of a specific outcome}[/tex]
  • [tex]n=\text{number of trials}[/tex]
  • [tex]p=\text{success rate}[/tex]
  • [tex]q=\text{failure rate, where }\boxed{q=1-p}[/tex]

Now, we put the given data into the formula:

[tex]\displaystyle \begin{aligned}P(X=5)&=\left(\begin{array}{c}10\\5\end{array}\right)(0.35)^5(1-0.35)^{10-5}\\\\&=\frac{10!}{5!(10-5)!} (0.35)^5(0.65)^5\\\\&=\frac{10!}{5!5!} (0.35)^5(0.65)^5\\\\&\bf\approx0.1546\end{aligned}[/tex]

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