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Sagot :
Given
A bag contains 13 red balls numbered 1-13 and 5 green balls numbered 14-18.
And a ball is chosen at random.
To find:
a) The probability that you choose a ren or even numbered ball.
b) The the probability you choose a green ball or a ball numbered less than 5.
Explanation:
It is given that,
The total number of balls is (13+5)=18.
The number of red balls is 13 numbered from 1-13.
The number of green balls is 5 numbered from 14-18.
i) Then, the probability of getting a a red or even numbered ball is,
[tex]\begin{gathered} P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right) \\ =\frac{n\left(A\right)}{n\left(S\right)}+\frac{n\left(B\right)}{n\left(S\right)}-\frac{n\left(A\cap B\right)}{n\left(S\right)} \end{gathered}[/tex]Here,
[tex]\begin{gathered} n\left(A\right)=n\left(red\text{ balls}\right) \\ =13 \\ n\left(B\right)=n\left(even\text{ numbered balls}\right) \\ =9 \\ n\left(A\cap B\right)=n\left(red\text{ balls and even numbered balls}\right) \\ =6 \\ n\left(S\right)=n\left(Total\text{ number of balls}\right) \\ =18 \end{gathered}[/tex]That implies,
[tex]\begin{gathered} P\left(A\cup B\right)=\frac{13}{18}+\frac{9}{18}-\frac{6}{18} \\ =\frac{16}{18} \\ =\frac{8}{9} \\ =0.89 \end{gathered}[/tex]Hence, the probability of getting a red or even numbered ball is 0.89.
ii) Also,
The probability of getting a green ball or a ball numbered less than 5 is,
[tex]\begin{gathered} P\left(C\cup D\right?=P\left(C\right)+P\left(D\right)-P\left(C\cap D\right) \\ =\frac{n\left(C\right)}{n\left(S\right)}+\frac{n\left(D\right)}{n\left(S\right)}-\frac{n\left(C\cap D\right)}{n\left(S\right)} \end{gathered}[/tex]Here,
[tex]\begin{gathered} n\left(C\right)=n\left(green\text{ balls}\right) \\ =5 \\ n\left(D\right)=n\left(balls\text{ numbered less than 5}\right) \\ =4 \\ n\left(C\cap D\right)=n\left(a\text{ }green\text{ }ball\text{ }or\text{ }a\text{ }ball\text{ }numbered\text{ }less\text{ }than\text{ }5\right) \\ =0 \end{gathered}[/tex]Then,
[tex]\begin{gathered} P\left(C\cup D\right)=\frac{5}{18}+\frac{4}{18}-0 \\ =\frac{9}{18} \\ =\frac{1}{9} \\ =0.11 \end{gathered}[/tex]Hence, the probabilty of getting a green ball or a ball numbered less than 5 is 0.11.
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