Answer: The probability that he pianist will be a boy and the alternate will be a girl is 30%.
Explanation:
Since we have given that
Number of boys are auditioning to play the piano for a school production = 2
Number of girls are auditioning to play the piano for a school production = 3
Total number of candidates is given by
[tex]2+3=5[/tex]
Now,
Let Event A denotes pianist will be a boy.
Let Event B denotes alternate will be girl.
Since Event A and B are independent events so,
[tex]P(\text{ A and B})=P(A)\times P(B)[/tex]
So,
Probability that the pianist will be a boy and the alternate will be a girl is given by
[tex]\frac{2}{5}\times \frac{3}{4}\\\\=\frac{6}{20}=\frac{3}{10}[/tex]
Now, we have to change it into percent ,
[tex]\frac{3}{10}\times 100=3\times 10=30\%[/tex]
Hence ,the probability that he pianist will be a boy and the alternate will be a girl is 30%.
The probability helps us to know the chances of an event occurring. The probability that the pianist will be a boy and the alternate will be a girl is 0.3.
The probability helps us to know the chances of an event occurring.
[tex]\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]
As there are 2 boys and 3 girls. And out of this total of 5 people, we need to know the probability of a boy being selected first. Therefore, the probability can be written as,
[tex]\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]
[tex]\rm Probability=\dfrac{\text{Number of boys}}{\text{Number of people}} = \dfrac{2}{5}[/tex]
Now since one boy is already selected the number of choices will reduce to 4, therefore, the probability of selecting a girl next can be written as,
[tex]\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]
[tex]\rm Probability=\dfrac{\text{Number of girls}}{\text{Number of people}} = \dfrac{3}{4}[/tex]
Thus, the probability that the pianist will be a boy and the alternate will be a girl can be written as,
[tex]\rm Probability=\dfrac25 \times \dfrac34 = \dfrac{3}{10} = 0.3[/tex]
Hence, the probability that the pianist will be a boy and the alternate will be a girl is 0.3.
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