Binomial Probability Mass Function

Contributions of Jacob Bernoulli to Probability

Jacob Bernoulli's Ars Conjectandi was published posthumously in $1713$. The book developed foundational results in probability, including Bernoulli's law of large numbers, and studied repeated success-or-failure trials that now bear his name.

Combination and Basic Probability

Before diving into binomial distribution, there are some basic formulas we need to learn first.

First, the combination formula for selecting objects:

Second, factorial (sequential multiplication):

By definition, $0!=1$.

And third, the relationship between event probability and its complement:

In this equation, $P(A)$ is the probability of an event occurring and $P(A^c)$ is the probability of the event not occurring.

Binomial Distribution Concept

A binomial random variable counts successes in repeated trials. Each trial has two mutually exclusive outcomes, such as heads or tails, and the label success identifies the outcome being counted.

The model applies only when the number of trials is fixed, the trials are independent, and the success probability stays constant. Independence means that an earlier result does not change the probability in a later trial.

Requirements for binomial experiments:

• There are only two possible outcomes: success or failure

• The number of experiments is predetermined and fixed

• Each experiment is mutually independent

• The probability of success is the same for each experiment

Binomial Probability Mass Function

For a binomial experiment with success probability $p$ and failure probability $q = 1-p$, the probability mass function gives the probability of exactly $x$ successes in $n$ independent trials:

Where:

• $n$ = fixed number of trials

• $x$ = number of successes being counted

• $p$ = success probability in one trial

• $q = 1-p$ = failure probability in one trial

Before applying the formula, verify all four binomial conditions. Otherwise, this probability model may not describe the experiment.

Coin Experiment

Look at a simple example. Suppose we have a fair coin. Its sample space contains heads and tails:

If we consider getting tails as "success", then the probability of success is $p = \frac{1}{2}$. Automatically, the probability of failure (getting heads) is $q = 1 - \frac{1}{2} = \frac{1}{2}$.

Problem: If we flip this coin $7$ times in a row, what's the probability of getting tails exactly $5$ times?

Solution:

First, identify the parameters:

• $n = 7$ (number of flips)
• $x = 5$ (number of tails desired)
• $p = \frac{1}{2}$ (probability of getting tails)
• $q = \frac{1}{2}$ (probability of getting heads)

Now use the binomial probability mass function:

Calculate $\binom{7}{5}$ first:

So the complete calculation is:

The probability of getting $5$ tails in $7$ flips is $\frac{21}{128}$ or approximately $16.4\%$.

Practice Problems

Each problem repeats one trial a fixed number of times and asks for an exact number of successes, so the binomial probability mass function applies directly. Identify the number of trials, the success probability, and the required number of successes before you compute.

1. In a bag there are $10$ balls that are exactly the same, except for their color: $6$ red balls and $4$ blue balls. If we draw $5$ balls one by one with replacement (each drawn ball is returned), what's the probability of drawing exactly $3$ red balls?

2. An archer has an accuracy rate of $80\%$ for hitting targets. If he takes $6$ shots, what's the probability he hits the target exactly $4$ times?

3. There's a multiple choice exam with $4$ answer choices for each question. A student guesses all answers randomly for $8$ questions. What's the probability he answers exactly $2$ questions correctly?

Answer Key

1. Red Ball Problem Answer

   Step 1: Identify parameters.

   • $n = 5$ (number of draws)
   • $x = 3$ (number of red balls desired)
   • $p = \frac{6}{10} = 0.6$ (probability of drawing a red ball)
   • $q = 1 - 0.6 = 0.4$ (probability of drawing a blue ball)

   Step 2: Calculate using the binomial probability mass function.

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

   Calculate $\binom{5}{3}$ first:

   $$\binom{5}{3} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10$$

   Then:

   $$= 10 \times (0.6)^3 \times (0.4)^2$$

   $$= 10 \times 0.216 \times 0.16 = 0.3456$$

   Therefore, the probability of drawing exactly $3$ red balls is $0.3456$ or approximately $34.6\%$.

2. Archer Problem Answer

   Step 1: Identify parameters.

   • $n = 6$ (number of shots)
   • $x = 4$ (number of targets desired)
   • $p = 0.8$ (probability of hitting target)
   • $q = 0.2$ (probability of missing)

   Step 2: Calculate using the binomial probability mass function.

   $$P(X=4) = \binom{6}{4} (0.8)^4 (0.2)^2$$

   Calculate $\binom{6}{4}$ first:

   $$\binom{6}{4} = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15$$

   Then:

   $$= 15 \times (0.8)^4 \times (0.2)^2$$

   $$\begin{aligned} &= 15 \times 0.4096 \times 0.04 \\ &= 0.24576 \\ &\approx 0.2458 \end{aligned}$$

   Therefore, the probability of the archer hitting the target exactly $4$ times is $0.2458$ or approximately $24.6\%$.

3. Multiple Choice Exam Problem Answer

   Step 1: Identify parameters.

   • $n = 8$ (number of questions)
   • $x = 2$ (number of correct answers desired)
   • $p = \frac{1}{4} = 0.25$ (probability of answering correctly by guessing)
   • $q = \frac{3}{4} = 0.75$ (probability of answering incorrectly)

   Step 2: Calculate using the binomial probability mass function.

   $$P(X=2) = \binom{8}{2} (0.25)^2 (0.75)^6$$

   Calculate $\binom{8}{2}$ first:

   $$\binom{8}{2} = \frac{8!}{2! \cdot 6!} = \frac{8 \times 7}{2 \times 1} = 28$$

   Then:

   $$= 28 \times (0.25)^2 \times (0.75)^6$$

   $$\begin{aligned} &= 28 \times 0.0625 \times 0.177978515625 \\ &= 0.31146240234375 \approx 0.3115\end{aligned}$$

   The probability of the student answering exactly $2$ questions correctly is approximately $0.3115$, or $31.15\%$.