Question

A database system assigns a 32-character ID to each record, where each character is either a number from 0 to 9 or a letter from A to F. Assume that each number or letter being selected is equally likely. Find the probability that at least 20 characters in the ID are numbers. Use Excel to find the probability. Round your answer to three decimal places.

Answer 1

Explanation:

total number of digits= 10 (from 0 to 9)

total number of letters = 6 (from A to F)

probability of numbers = 10/(10+6)

= 0.625

this is a case of binomial distribution with fixed number trials

n = 32 and probability p = 0.625

we have to find probability of at least 20 numbers

Use the BINOM.DIST function in Excel to find the cumulative probability.

P(at least 20 numbers) = 1 - BINOMDIST(numbers, trials, probability,true)

setting numbers = 20-1, trials = 32 and probability = 0.625

we get

`P(X \geq 20)=1 - BINOMDIST(20- 1, 32, 0.625, true) \\\\=1 -0.4219 \\\\=0.5781`

Alternatively,

The probability that there are exactly r letters can be found with binomial probability.

P = nCr pʳ qⁿ⁻ʳ

Given that n = 32, p = 5/8, and q = 3/8, you can use Excel to find each probability from r=20 to r=32, then add them all up.

P = ₃₂C₂₀ (⅝)²⁰ (⅜)³²⁻²⁰ + ₃₂C₂₁ (⅝)²¹ (⅜)³²⁻²¹ + ... + ₃₂C₃₂ (⅝)³² (⅜)³²⁻³²

P = 0.578