Step 1: Determine Total Outcomes
When a fair coin is tossed n times, each toss has 2 possibilities (Head or Tail).
Total number of possible outcomes = 2n
Step 2: Find the number of favorable outcomes (Odd Heads)
The number of ways to get exactly r heads in n tosses is given by the binomial coefficient nCr.
To get an odd number of heads, r can be 1,3,5,….
The sum of ways is:
Favorable ways=nC1+nC3+nC5+…
Step 3: Use Binomial Properties
We know from the properties of binomial coefficients that the sum of odd-positioned coefficients is equal to the sum of even-positioned coefficients:
nC0+nC2+nC4+⋯=nC1+nC3+nC5+⋯=2n−1
So, the number of ways to get an odd number of heads is 2n−1.
Step 4: Calculate the Probability
Probability=Total OutcomesFavorable Outcomes
P(Odd Heads)=2n−1⋅212n−1
Conclusion:
Regardless of the number of tosses n, the probability of getting an odd number of heads is always 21.
Correct Option: (a)
