Question

A bag contains $ 8 $ apple and $ 6 $ orange. Four fruits are drawn out one by one and not replaced. What is the probability that they are alternatively of different fruits?

Answer 1

Hint: Probability is the state of being probable and the extent to which something is likely to happen in the particular situations or the favourable outcomes. Probability of any given event is equal to the ratio of the favourable outcomes with the total number of the outcomes.
 $ P(A) = $ Total number of the favourable outcomes / Total number of the outcomes
Here, we will use the product of the probability when the total number of fruits and with the count of the specific favourable outcomes.

Complete step-by-step answer:
Four fruits are drawn out one by one and not replaced, and therefore can be two ways of drawing fruits, where it can be-
First way - AOAO and the second way - OAOA
O- An orange
A-An Apple
The total number of fruits is equal to the sum of the number of apples and the number of oranges.
Total number of fruits $ = 8 + 6 $
Total number of fruits $ = 14{\text{ }}.....{\text{ (i)}} $
The probability of the first way (AOAO) $ = \dfrac{8}{{14}} \times \dfrac{6}{{13}} \times \dfrac{7}{{12}} \times \dfrac{5}{{11}} $
(Here, we have used the ratio of the favourable outcomes to the total possible outcomes, without replacement and so we have reduced one-one count from the denominator in the product and same in the numerator part one is reduced from apple and orange)
Multiply the numerator and the denominator and simplify the above equation –
The probability of the first way (AOAO) $ = \dfrac{{10}}{{143}}\;{\text{ }}.....{\text{ (A)}} $
Similarly, the probability of the second way (OAOA) $ = \dfrac{6}{{14}} \times \dfrac{8}{{13}} \times \dfrac{5}{{12}} \times \dfrac{7}{{11}} $
Multiply the numerator and the denominator and simplify the above equation –
The probability of the second way (OAOA) $ = \dfrac{{10}}{{143}}\;{\text{ }}....{\text{ (B)}} $
Now, the total probability by using the equations (A) and (B)
The total probability $ = \dfrac{{10}}{{143}} + \dfrac{{10}}{{143}} $
When the denominators are the same, add the numerators directly.
 The total probability $ = \dfrac{{20}}{{143}} $
So, the correct answer is “ $ \dfrac{{20}}{{143}} $ ”.

Note: Remember to reduce the count of the total number of possible outcomes, when the things are not replaced and withdrawn from any box or bag. The probability of any event always ranges between zero and one. It can never be the negative number or the number greater than one.