2.4.2 Binomial distribution

    Cards (99)

    • The Normal distribution is used for continuous data, while the Binomial distribution is used for discrete data.
    • The formula for the probability of exactly kk successes in nn trials is P(X=k)=P(X = k) =(nk)pk(1p)nk \binom{n}{k} p^{k} (1 - p)^{n - k}, where (nk)\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 nn.
    • 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 kk successes in nn trials is P(X=k)=P(X = k) =(nk)pk(1p)nk \binom{n}{k} p^{k} (1 - p)^{n - k}, where (nk)\binom{n}{k} is the binomial coefficient
    • How is the binomial coefficient (nk)\binom{n}{k} calculated?

      n!k!(nk)!\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 (nk)\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 pp and must remain constant across all trials.
    • What is the formula for calculating the probability of exactly kk successes in nn trials in a Binomial distribution?

      P(X=k)=P(X = k) =(nk)pk(1p)nk \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 (nk)\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 (nn), number of successes (kk), and probability of success (pp).
    • 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)=P(X = k) =(nk)pk(1p)nk \binom{n}{k} p^{k} (1 - p)^{n - k}
    • P(X=k)P(X = k) represents the probability of exactly kk successes.
    • The binomial coefficient (nk)\binom{n}{k} represents the number of ways to choose kk successes from nn trials
    • What does (1p)nk(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=n =5 5, k=k =3 3, p=p =0.7 0.7
      2️⃣ Calculate binomial coefficient: (53)=\binom{5}{3} =10 10
      3️⃣ Substitute into binomial formula: P(X=3)=P(X = 3) =10×(0.7)3×(0.3)2 10 \times (0.7)^{3} \times (0.3)^{2}
      4️⃣ Calculate probability: P(X=3)=P(X = 3) =0.3087 0.3087
    • What is the binomial coefficient (53)\binom{5}{3}?

      1010
    • 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