Question

A fair coin is tossed 100 times. The Probability of getting 50 heads is close to one of the following numbers:

a)0.001

b)0.1

c)0.3  

d)0.4

Answer 1

Assuming we need exactly 50 heads

¹⁰⁰C₅₀ 1/2⁵⁰ 1/2⁵⁰ ≈ 0.1

Comments

This conventional way can't be calculated on calculator.
Is there smart trick to calculate the formula easily?

---

I did on calculator only - you can use scientific one so large numbers will be processed as double which loose precision but here thats not a problem.

---

Thanks.
I'll use scientific one next time.

Answer 2

The probability of getting exactly k heads in n tosses of a fair coin is given by the binomial distribution formula:

P(k heads in n tosses) = (n choose k) * p^k * (1-p)^(n-k)

where p is the probability of getting a head on any one toss of the coin.

In this case, we want to find the probability of getting 50 heads in 100 tosses of a fair coin, so we have:

P(50 heads in 100 tosses) = (100 choose 50) * 0.5^50 * 0.5^50

= (100 choose 50) * 0.5^100

Using Stirling's approximation for factorials, we can approximate (100 choose 50) as:

(100 choose 50) ≈ (2pi50)^(-1/2) * (100/50)^50

Plugging this into the expression above, we get:

P(50 heads in 100 tosses) ≈ (2pi50)^(-1/2) * (100/50)^50 * 0.5^100

≈ 0.0796

So the closest answer choice is 0.1.