UPSC NDA I 2024 Mathematics Paper

To solve for the value of n, we need to use the given probabilities and the fact that the coin is unbiased, meaning the probability of getting heads (H) and tails (T) are both ‌

1

2

.
The probability of getting at least one tail in n coin tosses is p. The complementary event, i.e., getting no tails (all heads), happens with a probability of (‌

1

2

)n.
Therefore, we can write the probability of getting at least one tail as:
p=1−(‌

1

2

)n
Now, for the probability of getting at least two tails (q), we consider the complementary events, which are getting zero tails or exactly one tail. The probability of getting exactly one tail in n coin tosses can be calculated using the binomial distribution:
(

n

1

)(‌

1

2

)1(‌

1

2

)n−1=n(‌

1

2

)n
Therefore, the probability of getting at least two tails is:
q=1−(‌

1

2

)n−n(‌

1

2

)n=1−(‌

1

2

)n(1+n)
We are given that p−q=‌

5

32

. Substituting the expressions for p and q, we get:
(1−(‌

1

2

)n)−(1−(‌

1

2

)n(1+n))=‌

5

32

Simplifying, we obtain:
1−(‌

1

2

)n−1+(‌

1

2

)n(1+n)=‌

5

32

(‌

1

2

)n(1+n)−(‌

1

2

)n=‌

5

32

(‌

1

2

)nn=‌

5

32

Solving for n, we look at reasonable values of n that make the equation true. Let's explore the given options:
For n=4 :
(‌

1

2

)4×4=‌

1

16

×4=‌

4

16

=‌

1

4

This does not match ‌

5

32

.
For n=5 :
(‌

1

2

)5×5=‌

1

32

×5=‌

5

32

This matches the given equation ‌

5

32

.
So, the correct value of n is 5 .