# If probability of success in a single trial is 0.01. How many trials are required in order to have probability greater than 0.5 of getting at least one success? - Mathematics and Statistics

Sum

If the probability of success in a single trial is 0.01. How many trials are required in order to have a probability greater than 0.5 of getting at least one success?

#### Solution

Let X = number of successes.

p = probability of success in a single trial

∴ p = 0.01

and q = 1 - p = 1 - 0.01 = 0.99

∴ X ~ B(n, 0.01)

The p.m.f. of X is given by

P(X = x) = "^nC_x  p^x q^(n - x)

i.e. p(x) = "^nC_x  (0.01)^x  (0.99)^(n - x)  x = 1, 2,....,n

P(at least one success)

= P(X ≥ 1) = 1 - P(X < 1)

= 1 - P(X = 0) = 1 - p(0)

= 1 - "^nC_0  (0.01)^0  (0.99)^(n - 0)

= 1 - 1(1)(0.99)^n

= 1 - (0.99)^n

Given: P(X ≥ 1) > 0.5

i.e. 1 - (0.99)^n > 0.5

i.e. 1 - 0.5 > (0.99)^"n"

i.e. 0.5 > (0.99)^"n"

i.e. (0.99)^"n" < 0.5

i.e. log (0.99)^"n" < log (0.5)

i.e. n log (0.99) < log 0.5

i.e. n < ("log"  0.5)/("log"  0.99)

i.e. n < 68.96

∴ n = 68

Hence, the number of trials required in order to have probability greater than 0.5 of getting at least one success is ("log"  0.5)/("log"  0.99) OR 68

Concept: Binomial Distribution
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 8 Binomial Distribution
Miscellaneous exercise 8 | Q 16 | Page 255
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