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 \[P\left( A\cap B \right)=0.72\].
\[\begin{align}
  & \therefore P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right) \\
 & P\left( A\cup B \right)=0.9+0.8-0.72=0.98 \\
\end{align}\]
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 \[\dfrac{2}{100}\].