Calculating Binomial Probability | A Mathematical Formula for Successes in Independent Trials

Binomial Probability Formula

The binomial probability formula is a mathematical formula that allows you to calculate the probability of a specific number of successes in a fixed number of independent Bernoulli trials

The binomial probability formula is a mathematical formula that allows you to calculate the probability of a specific number of successes in a fixed number of independent Bernoulli trials.

The formula is given by:

P(x) = (nCx) * p^x * (1-p)^(n-x)

Where:
– P(x) represents the probability of exactly x successes
– n is the total number of trials or attempts
– x is the number of successes you are interested in
– (nCx) is the number of ways to choose x items out of n items and can be calculated using the combination formula (nCx) = n! / (x!(n-x)!), where “!” denotes a factorial
– p is the probability of success on each trial

In simpler terms, the formula represents the probability of getting exactly x successes in n independent trials, with each trial having a probability of p for success and (1-p) for failure.

For example, let’s say you are flipping a fair coin 5 times (n = 5) and you want to find the probability of getting exactly 3 heads (x = 3). Since the coin is fair, the probability of getting a head in any single flip is 0.5 (p = 0.5). Using the binomial probability formula:

P(3) = (5C3) * (0.5)^3 * (0.5)^(5-3)
= (5! / (3!(5-3)!)) * (0.5)^3 * (0.5)^2
= (5! / (3!*2!)) * (0.5)^3 * (0.5)^2
= (5 * 4 * 3! / (3!*2 * 1)) * 0.125 * 0.25
= (5 * 4 / 2) * 0.03125
= 10 * 0.03125
= 0.3125

Therefore, the probability of getting exactly 3 heads when flipping a fair coin 5 times is 0.3125 or 31.25%.

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