2.4.2 Binomial distribution

• The Normal distribution is used for continuous data, while the Binomial distribution is used for discrete data.
• The formula for the probability of exactly $k$ successes in $n$ trials is $P(X = k) =$$\binom{n}{k} p^{k} (1 - p)^{n - k}$, where $\binom{n}{k}$ is the binomial coefficient
• The Binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials
• In a Binomial distribution, the number of trials is fixed and denoted as $n$.
• In a Binomial distribution, each trial is independent, meaning the outcome of one trial does not affect the outcome of others
• The probability of success in a Binomial distribution must remain constant across all trials.
• The formula for calculating the probability of exactly $k$ successes in $n$ trials is $P(X = k) =$$\binom{n}{k} p^{k} (1 - p)^{n - k}$, where $\binom{n}{k}$ is the binomial coefficient
• How is the binomial coefficient $\binom{n}{k}$ calculated?


  $\frac{n!}{k!(n - k)!}$
• The Binomial distribution is a discrete probability distribution for successes in independent trials
• In a Binomial distribution, the outcome of each trial does not affect subsequent trials.
• The Binomial distribution differs from the Poisson distribution in terms of the number of possible outcomes
• What does the binomial coefficient $\binom{n}{k}$ represent in the Binomial distribution formula?


  Number of ways to choose successes
• In a Binomial distribution, the probability of success must remain the same for each trial
• The Binomial distribution models the number of successes in a fixed number of independent trials.
• In a Binomial distribution, each trial must be independent of the others
• The probability of success in a Binomial distribution is denoted as $p$ and must remain constant across all trials.
• What is the formula for calculating the probability of exactly $k$ successes in $n$ trials in a Binomial distribution?


  $P(X = k) =$$\binom{n}{k} p^{k} (1 - p)^{n - k}$
• The Binomial distribution is a discrete probability distribution for independent trials with two possible outcomes
• Independent trials in a Binomial distribution mean the outcome of one trial affects the outcome of others.

  False
• In a Binomial distribution, the number of trials must be a fixed value.
• The probability of success in a Binomial distribution remains constant across all trials.
• What does the binomial coefficient $\binom{n}{k}$ in the Binomial distribution formula represent?


  Number of ways to choose successes
• The Binomial distribution is a discrete probability distribution for independent trials with two possible outcomes
• In a Binomial distribution, the outcome of one trial does not affect the outcome of others.
• Match the distribution with its conditions:

  Binomial ↔️ Fixed trials, independent events

  Poisson ↔️ Events occurring randomly

  Normal ↔️ Continuous data
• In a Binomial distribution, trials must be independent.
• What are the two possible outcomes for each trial in a Binomial distribution?

  Success or failure
• The probability of success in a Binomial distribution is denoted asp</latex> and must remain constant across all trials.
• To calculate probabilities using the Binomial Distribution formula, the first step is to identify the number of trials ($n$), number of successes ($k$), and probability of success ($p$).
• What is the first step in calculating probabilities using the Binomial Distribution formula?

  Identify the parameters
• What is the Binomial probability formula?

  $P(X = k) =$$\binom{n}{k} p^{k} (1 - p)^{n - k}$
• $P(X = k)$ represents the probability of exactly $k$ successes.
• The binomial coefficient $\binom{n}{k}$ represents the number of ways to choose $k$ successes from $n$ trials
• What does $(1 - p)^{n - k}$ in the Binomial formula represent?


  Probability of failure raised to the number of failures
• Steps to calculate the probability of making exactly 3 out of 5 free throws, given a success probability of 0.7

  1️⃣ Identify parameters: $n =$$5$, $k =$$3$, $p =$$0.7$

  2️⃣ Calculate binomial coefficient: $\binom{5}{3} =$$10$

  3️⃣ Substitute into binomial formula: $P(X = 3) =$$10 \times (0.7)^{3} \times (0.3)^{2}$

  4️⃣ Calculate probability: $P(X = 3) =$$0.3087$
• What is the binomial coefficient $\binom{5}{3}$?


  $10$
• The probability of making exactly 3 out of 5 free throws is 0.3087.
• The Binomial Distribution is a discrete probability distribution used to model the number of successes in a fixed number of independent trials
• What are the four conditions for a Binomial Distribution?

  Fixed trials, independent events, two outcomes, constant success probability
• Match the distribution with its conditions:

  Binomial ↔️ Fixed trials, independent events

  Poisson ↔️ Events occurring randomly

  Normal ↔️ Continuous data