Answer :

MathPhys

Answer:

a) 5/16

b) 31/32

Step-by-step explanation:

A coin toss is an example of a binomial distribution, where the following conditions are met:

  • A discrete and fixed number of trials, n
  • Trials are independent of each other
  • Two possible outcomes: success and failure
  • Probability of success, p, is constant

In this case, there are 5 coin tosses, so n = 5. There are two outcomes, heads (success) and tails (failure). Each coin toss is independent, and the probability of success is always p = 0.5.

The probability of getting exactly r number of successes out of n trials is:

P = nCr pʳ (1−p)ⁿ⁻ʳ

where

  • nCr is the number of combinations of r successes from n trials
  • p is the probability of success
  • r is the number of successes
  • 1−p is the probability of failure
  • n−r is the number of failures

Since n = 5 and p = 0.5, this equation becomes:

P = 5Cr (0.5)ʳ (1−0.5)⁵⁻ʳ

P = 5Cr (0.5)ʳ (0.5)⁵⁻ʳ

P = 5Cr (0.5)⁵

a) If r = 3, then the probability is:

P = ₅C₃ (0.5)⁵

P = 10 (0.5)⁵

P = 5/16

b) The probability of at least one head is the probability of one head, two heads, three heads, four heads, or five heads.

P(at least 1) = P(1) + P(2) + P(3) + P(4) + P(5)

Or, we can simply say it is 1 minus the probability of no heads.

P(at least 1) = 1 − P(0)

P(at least 1) = 1 − ₅C₀ (0.5)⁵

P(at least 1) = 1 − 1 (0.5)⁵

P(at least 1) = 31/32

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