smileycellist2

2021-09-19

Suppose that a medicine cures 72% of people with a particular disease. If 7 people are sick, what is the probability that at least six of them will be cured by this drug? Explain your reasoning.

Margot Mill

Step 1- Writing given data
Given that
A medicine cures 72% of people with a particular disease .
$P=0.72$
sample size $\left(n\right)=7$
We know that
Binomial probability distribution
$P\left(X=x\right){=}^{n}{C}_{x}x{P}^{x}x{\left(1-P\right)}^{\left(n-x\right)}$
Step 2 - Probability calculation
What is the probability that at least six of them will be cured by this drug $=P\left(X\ge 6\right)$
$P\left(X\ge 6\right)=P\left(X=6\right)+P\left(X=7\right)$
Here,
$n=7;P=0.72;\left(1-P\right)=0.28$
$P\left(X\ge 6\right){=}^{7}{C}_{6}x{0.72}^{6}x{\left(0.28\right)}^{\left(7-6\right)}{+}^{7}{C}_{7}x{0.72}^{7}x{\left(0.28\right)}^{\left(7-7\right)}$
$P\left(X\ge 6\right)=\left[7x{0.72}^{6}x{0.28}^{1}\right]+\left[1x{0.72}^{7}x1\right]$
$P\left(X\ge 6\right)=0.273055+0.100306$
$P\left(X\ge 6\right)=0.373361$
The probability that at least six of them will be cured by this drug is 0.3734
Here,
Binomial probability distribution is used because here we have only two chances, curing or not curing.

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