The binomial and geometric distributions – The binomial distribution

A binomial experiment has four conditions:

• A fixed number of trials \(n\)
• Each trial has two possible outcomes (success or failure)
• The probability of success \(p\) is constant
• Trials are independent

If a random variable \(X\) counts the number of successes:

\(X \sim B(n,p)\)

The probability of exactly \(r\) successes is:

P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}

• \(n\) = number of trials
• \(r\) = number of successes
• \(p\) = probability of success

If \(X \sim B(n,p)\):

• Mean: \( \mu = np \)
• Variance: \( \sigma^2 = np(1-p) \)
• Standard deviation: \( \sigma = \sqrt{np(1-p)} \)

A biased coin has probability \(p=0.6\) of landing heads.

The coin is tossed \(n=5\) times.

Let \(X\) be the number of heads.

\(X \sim B(5,0.6)\)

Find \(P(X=3)\).

Step 1: Substitute into formula

\[ P(X=3) = \binom{5}{3}(0.6)^3(0.4)^2 \]

Step 2: Calculate

\[ \binom{5}{3} = 10 \] \[ P(X=3) = 10(0.216)(0.16) \]

If \(X \sim B(20,0.3)\):

Mean:

\[ \mu = np = 20(0.3) = 6 \]

Variance:

\[ \sigma^2 = np(1-p) = 20(0.3)(0.7) = 4.2 \]

Standard deviation:

\[ \sigma = \sqrt{4.2} \approx 2.05 \]

• Always define the random variable clearly.
• Write the distribution: \(X \sim B(n,p)\).
• Use the correct binomial formula or calculator.
• Remember that \(q = 1-p\).
• Check that trials are independent before using a binomial model.