Answer to Question #88207 in Statistics and Probability for Marine Smit

Question #88207
It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers:
1.1 At least 2 will have a medical aid
1.2. More than 9 will have a medical aid
1
Expert's answer
2019-04-18T02:56:03-0400

The binomial probability formula is the result of getting exactly k successes in n number of trials.


P(k success in n trials)=(nk)pk(1p)nkP(k \ success\ in\ n \ trials)=\begin{pmatrix} n \\ k \end{pmatrix}p^k(1-p)^{n-k}

We have that


n=10,p=0.3n=10, p=0.3

1.1 At least 2 will have a medical aid 


P(X2)=1(P(X=0)+P(X=1))=P(X\geq 2)=1-(P(X=0)+P(X=1))=

=1(100)0.30(10.3)100(101)0.31(10.3)1010.850692=1-\begin{pmatrix} 10 \\ 0 \end{pmatrix}0.3^0(1-0.3)^{10-0}-\begin{pmatrix} 10 \\ 1 \end{pmatrix}0.3^1(1-0.3)^{10-1}\approx 0.850692

1.2. More than 9 will have a medical aid


P(X>9)=P(X=10)=(1010)0.310(10.3)10100.000006P(X> 9)=P(X=10)=\begin{pmatrix} 10 \\ 10 \end{pmatrix}0.3^{10}(1-0.3)^{10-10}\approx 0.000006

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