{"id":842015,"date":"2024-09-11T13:44:15","date_gmt":"2024-09-11T08:14:15","guid":{"rendered":"https:\/\/leverageedu.com\/discover\/?p=842015"},"modified":"2024-09-11T13:44:15","modified_gmt":"2024-09-11T08:14:15","slug":"exam-prep-binomial-distribution","status":"publish","type":"post","link":"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-binomial-distribution\/","title":{"rendered":"Binomial Distribution: Definition, Properties and Solved Examples"},"content":{"rendered":"\n

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is defined by two parameters: the number of trials (n) and the probability of success in each trial (p). The formula to calculate the probability of exactly k successes is P(X=k) = (n<\/sup>k<\/sub>)pk<\/sup>(1\u2212p)n-k<\/sup><\/strong>. Key properties include mean (np) and variance (npq, where q = 1 – p). This distribution is fundamental for analyzing binary outcomes and is frequently tested in competitive exams like IIT-JEE<\/a><\/strong>, GATE<\/strong>, GRE<\/a><\/strong>, and CAT<\/a><\/strong>, where solving related problems helps understand real-life applications like quality control and decision-making under uncertainty.<\/p>\n\n\n\n\n\n\n

What is Binomial Distribution?<\/h2>\n\n\n\n

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent and identical trials, where each trial has only two possible outcomes: success or failure. The probability of success remains constant across trials, and each trial is independent of others. The distribution is used to calculate the likelihood of achieving a specific number of successes in a given set of trials. Important terms related to binomial distribution are:<\/p>\n\n\n\n

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  1. Trial<\/strong>: A single event with two possible outcomes (success\/failure).<\/li>\n\n\n\n
  2. Success (p)<\/strong>: The probability of the desired outcome in a single trial.<\/li>\n\n\n\n
  3. Failure (q)<\/strong>: The probability of the undesired outcome, where q=1\u2212p.<\/li>\n\n\n\n
  4. Number of Trials (n)<\/strong>: The total number of independent trials or experiments conducted.<\/li>\n\n\n\n
  5. Number of Successes (k)<\/strong>: The number of successful outcomes in the trials.<\/li>\n\n\n\n
  6. Binomial Coefficient<\/strong> (<\/strong>n<\/sup><\/strong>k<\/sub><\/strong>)<\/strong>: A combination that calculates the number of ways to choose k successes from n trials, represented as n!\/k!(n\u2212k)!<\/strong>\u200b.<\/li>\n\n\n\n
  7. Probability Mass Function (PMF)<\/strong>: The formula used to calculate the probability of exactly k successes: P(X=k) = (<\/strong>n<\/sup><\/strong>k<\/sub><\/strong>)p<\/strong>k<\/sup><\/strong>(1\u2212p)<\/strong>n-k<\/sup><\/strong>.<\/li>\n\n\n\n
  8. Mean (np)<\/strong>: The expected number of successes in n trials.<\/li>\n\n\n\n
  9. Variance (npq)<\/strong>: A measure of the spread or variability in the number of successes.<\/li>\n<\/ol>\n\n\n\n

    Must Read: Commercial Maths<\/a><\/strong><\/p>\n\n\n\n

    Properties of Binomial Distribution<\/h2>\n\n\n\n

    The binomial distribution<\/strong> has several key properties that describe its behavior and characteristics. These properties are crucial in understanding the distribution’s shape, central tendency, and variability<\/p>\n\n\n\n

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    1. Two Possible Outcomes<\/strong>: Each trial has only two possible outcomes: success (with probability p) and failure (with probability q=1\u2212p).<\/li>\n\n\n\n
    2. Fixed Number of Trials (n)<\/strong>: The number of trials, n, is fixed and predetermined. Each trial is independent of others, meaning the outcome of one trial does not affect the others.<\/li>\n\n\n\n
    3. Constant Probability (p)<\/strong>: The probability of success, p, remains the same for all trials.<\/li>\n\n\n\n
    4. Discrete Distribution<\/strong>: The binomial distribution is discrete, meaning it takes only integer values (the number of successes) ranging from 0 to n.<\/li>\n\n\n\n
    5. Mean (Expected Value)<\/strong>: The mean or expected value of the binomial distribution is given by:
      \u03bc = np<\/strong>
      This represents the average number of successes expected in n trials.<\/li>\n\n\n\n
    6. Variance<\/strong>: The variance, which measures the spread of the distribution, is:
      \u03c32 <\/sup>= npq<\/strong>
      where q = 1\u2212p is the probability of failure. This shows how much the number of successes is expected to deviate from the mean.<\/li>\n\n\n\n
    7. Standard Deviation<\/strong>: The standard deviation is the square root of the variance: \u03c3 = \u221anpq<\/strong><\/li>\n\n\n\n
    8. Symmetry and Skewness<\/strong>: The binomial distribution becomes more symmetric as n increases and when p is close to 0.5. If p is much greater than 0.5, the distribution is skewed to the left (negatively skewed), while if p is much less than 0.5, it is skewed to the right (positively skewed).<\/li>\n\n\n\n
    9. Maximum Probability<\/strong>: The most probable value (the mode) of the binomial distribution is around k = np (the mean), but it may vary slightly depending on whether p is closer to 0 or 1.<\/li>\n\n\n\n
    10. Relationship with Normal Distribution<\/strong>: For large n, the binomial distribution approaches a normal distribution, especially when p is close to 0.5. This is known as the normal approximation<\/strong> to the binomial distribution, often applied when np \u2265 5 and nq \u2265 5.<\/li>\n<\/ol>\n\n\n\n

      Formula of Binomial Distribution<\/h2>\n\n\n\n
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