Question

The probability of a boy student getting scholarship is 0.9 and that of a girl student getting scholarship is 0.8. The probability that at least one of them will get scholarship is _____
(a). $\dfrac{98}{100}$
(b). $\dfrac{2}{100}$
(c). $\dfrac{72}{100}$
(d).$\dfrac{28}{100}$

Answer 1

Hint: Take the probability of boy and girl students getting scholarships as P (A) and P (B). Thus find $P\left( A\cap B \right)$ as both A and B are independent events. The probability of at least one of them can also mean that they both get the scholarship. So find $P\left( A\cup B \right)$.

Complete step-by-step solution -

It is said that the probability of a boy getting a scholarship is 0.9.
Let A be the event that a boy receives the scholarship. Thus we can write that P (A) = 0.9.
The probability that a girl will get a scholarship is 0.8. Let B be the event that a girl receives a scholarship.
$\therefore$ P (B) = 0.8
A and B, both are independent events. So we need to find $P\left( A\cap B \right)$.
Now, $P\left( A\cap B \right)$ = P (A). P (B)
We know, P (A) = 0.9 and P (B) = 0.8.
$\therefore P\left( A\cap B \right)=0.9\times 0.8=0.72$
We need to find the probability that at least one of them will get a scholarship, which means there are chances that both the boy and girl will get a scholarship. Thus we need to find, $P\left( A\cup B \right)$.
i.e. Probability that at least one of them will get scholarship = $P\left( A\cup B \right)$
We know, $P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$.
Now let us substitute the values and simplify it, P (A) = 0.9, P (B) = 0.8 and $P\left( A\cap B \right)=0.72$.

& \therefore P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right) \\\ & P\left( A\cup B \right)=0.9+0.8-0.72=0.98 \\\ \end{aligned}$$ Thus we got the probability that at least one of them will get as 0.98, which can be written as $$\dfrac{98}{100}$$ in fractional form. Thus we got the probability as $$\dfrac{98}{100}$$. $$\therefore $$Option (a) is the correct answer. Note: If we were asked to find the probability that none of them will get the scholarship then: $$1-P\left( A\cup B \right)$$ will give the probability. $$\therefore $$ Probability that none got scholarship = $$1-P\left( A\cup B \right)$$ $$=1-\dfrac{98}{100}=\dfrac{100-98}{100}=\dfrac{2}{100}=0.02$$ Thus the probability that none will get are 0.02 or $$\dfrac{2}{100}$$.