Probability using Binomial Distribution

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards; find the probability that none is a spade.

The probability that a lamp in a classroom will be burnt out is 0.3. Six such lamps are fitted in the class-room. If it is known that the classroom is unusable if the number of lamps burning in it is less than four, find the probability that the classroom cannot be used on a random occasion.

The probability that a machine will produce all bolts in a production run within specification is 0.998. A sample of 8 machines is taken at random. Calculate the probability that all 8 machines.

The probability that a machine will produce all bolts in a production run within specification is 0.998. A sample of 8 machines is taken at random. Calculate the probability that 7 or 8 machines.

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at most 5 successes.

Let X ~ B(10, 0.2). Find P(X ≤ 8).

If the mean and variance of a binomial distribution are 18 and 12 respectively, then n = ______.

A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of getting at least two success.

Solution:

A pair of dice is thrown 3 times.

∴ n = 3

Let x = number of success (doublets)

p = probability of success (doublets)

∴  p = `square`, q = `square`

∴ x ∼ B (n, p)

P(x) = ⁿC_xp^x q^n–x

Probability of getting at least two success means x ≥ 2.

∴ P(x ≥ 2) = P(x = 2) + P(x = 3)

= `square` + `square`

= `2/27`