Binomial probability helps us calculate the chances of specific outcomes in experiments with fixed trials and two possible results. It's like predicting how many times you'll roll a six in ten dice throws.

The binomial formula is key, using factors like total trials, desired successes, and individual success probability. We can use this to figure out things like the likelihood of getting three heads when flipping a coin five times.

Binomial Probability

Binomial formula for probabilities

• Calculates the probability of a specific number of successes in a fixed number of independent trials
• Formula: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$ (also known as the probability mass function for discrete probability distributions)
  • $n$: total number of trials (fixed)
  • $k$: number of successes desired
  • $p$: probability of success on a single trial
  • $1-p$: probability of failure on a single trial
• To use the formula:
  1. Confirm the experiment is binomial (fixed trials, two outcomes, constant probability, independence)
  2. Identify $n$, $k$, and $p$ based on the problem
  3. Calculate the combination $\binom{n}{k}$ using $\frac{n!}{k!(n-k)!}$
  4. Plug values into the formula and simplify
• Examples:
  • Probability of getting exactly 3 heads in 5 coin flips (fair coin)
  • Probability of 2 defective items in a batch of 10 (10% defect rate)

Probability functions for binomial experiments

• Probability density function (PDF) gives the probability of each possible outcome
  • Construct by calculating $P(X=k)$ for each $k$ from 0 to $n$ using the binomial formula
  • All probabilities in a PDF sum to 1
  • Example: PDF for number of heads in 3 coin flips (0, 1, 2, or 3 heads possible)
• Cumulative distribution function (CDF) gives the probability of $k$ or fewer successes
  • Construct by summing probabilities from the PDF for each $k$ from 0 to the desired value
  • CDF starts at 0 and ends at 1
  • Example: CDF for 2 or fewer defective items in a batch of 10

Binomial Experiments

Criteria of binomial experiments

• Fixed number of trials ($n$)
• Only two possible outcomes per trial (success or failure)
• Constant probability of success ($p$) for all trials
• Trials are independent of each other (independence ensures that the outcome of one trial does not affect the others)
• Examples of binomial experiments:
  • Flipping a coin 20 times and counting heads (two outcomes, fixed trials, constant $p$, independence)
  • Testing 15 batteries and counting defects (pass/fail, fixed trials, constant defect rate, independence)
• Non-examples:
  • Drawing cards without replacement (probability changes each draw, violating constant $p$)
  • Survey with multiple choice answers (more than two outcomes per trial)

Properties of Binomial Distribution

• Mean (expected value): $\mu = np$
• Variance: $\sigma^2 = np(1-p)$, which measures the spread of the distribution
• The binomial distribution is a discrete probability distribution, meaning it deals with countable, distinct outcomes