Binomial distribution

Binomial distribution


P(x)=nCxPx(1p)nxP(x)={_n}C_x \;P^x(1-p)^{n-x}

nn: number of trials
xx: number of success in n trials
pp: probability of success in each trial
P(x)P(x): probability of getting xx successes (out of nn trials)

\cdot binomialpdf (n,p,x)(n,p,x)
\cdot binomialcdf (n,p,x)(n,p,x)
  • 1.

    Binomial Formula

    Binomialpdf Calculator

  • 2.
    Identify which of the following experiments below are binomial distributions?

    i. A fair die is rolled 4 times. What is the probability of the one coming up 2 times?

    ii. A fair coin is flipped until head comes up 7 times. What is the probability that the coin will be flipped 10 times?

    iii. 1,000,000 nails are produced in a factory a day. If each nail has a probability of 0.5% of being defective (something being wrong with that nail), then what is the probability that less than 50 nails will be defective in a day?

    iv. Roughly 7.5% of Canadians have some form of heart disease. If 100 Canadians are sampled what is the probability that 10 of them will have heart disease?

    v. If 5 cards are drawn from a deck, what is the probability that 2 of them will be hearts?

    vi. If a fair die is rolled 8 times, what is the probability of getting 2 fours and 3 sixes?

  • 3.
    An urn contains 6 red balls and 4 green balls. A total of 5 balls are drawn; list all the different combinations of red balls that can be drawn in each of the following cases:

    i. A total of 3 green balls are drawn

    ii. At most 3 red balls are drawn

    iii. At least 2 red balls are drawn

    iv. Less than 4 red balls are drawn

    v. More than 3 green balls are drawn

  • 4.
    A die is rolled 3 times, what is the probability that a four is rolled exactly 2 times?

  • 5.
    A coin is flipped 20 times, what is the probability that the coin comes up heads 15 times?

  • 6.
    Jimmy the Joker is an unfair gambler. He weights a die so it rolls a “6” with 75% chance. He then bets that if he rolls his die 4 times he will roll six exactly 3 times. What is his probability of winning this bet?

  • 7.
    Thomas is packing for a trip and wants to bring some stuffed animals along for comfort. He owns 8 stuffed animals, and will pack each stuffed animals independently of all the others with a probability of 30%. Determine the probability that he takes;
    0 stuffed animals along with him.

    1 stuffed animal with him

    at most two animals along with him.

    at most 5 animals along with him.

    at least 6 animals along with him.