Question

For the following question, enter the correct numerical value upto TWO decimal places. If the numerical value has more than two decimal places, round-off the value to TWO decimal places. (For example: Numeric value 5 will be written as 5.00 and 2.346 will be written as 2.35) Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is ____.

Hint:

For repeated independent trials, \[ P(\text{at least one success}) = 1 - P(\text{no success}) \] Use logarithms to solve exponential probability inequalities.

Answer 1

Correct Answer: 7

Step 1: Let the coin be tossed \( n \) times. The probability of getting no head (i.e., all tails) is: \[ \left(\frac{1}{2}\right)^n \]
Step 2: Therefore, the probability of getting at least one head is: \[ 1 - \left(\frac{1}{2}\right)^n \]
Step 3: Given that this probability is more than \(99% = 0.99\): \[ 1 - \left(\frac{1}{2}\right)^n>0.99 \]
Step 4: Rearranging: \[ \left(\frac{1}{2}\right)^n<0.01 \] Taking logarithms: \[ n \log\left(\frac{1}{2}\right)<\log(0.01) \]
Step 5: Using logarithmic values: \[ n>\frac{\log(0.01)}{\log(0.5)} = \frac{-2}{-0.3010} \approx 6.64 \]
Step 6: Since \( n \) must be a whole number, the minimum value satisfying the condition is: \[ n = 7 \] Final Answer (up to two decimal places): \[ \boxed{7.00} \]