How can you tell when you need to use the binomial coefficient for probabilities?For example, the probability of being dealt a hand with 3 spades and 2 hearts from a standard deck of cards. I would think the geometric probability is: 13 ∗ 12 ∗ 11 ∗ 13 ∗ 12 52 ∗ 51 ∗ 50 ∗ 49 ∗ 48 because I usually don't use the binomial coefficient with geometric probabilities, but the answer is incorrect unless I multiply by the binomial coefficient (which is a factor of 10 here).

Question

How can you tell when you need to use the binomial coefficient for probabilities?For example, the probability of being dealt a hand with 3 spades and 2 hearts from a standard deck of cards. I would think the geometric probability is:                    13        ∗        12        ∗        11        ∗        13        ∗        12                    52        ∗        51        ∗        50        ∗        49        ∗        48             because I usually don&#039;t use the binomial coefficient with geometric probabilities, but the answer is incorrect unless I multiply by the binomial coefficient (which is a factor of 10 here).

Kason Gonzales · Accepted Answer

Step 1If X is a binomial random variable with parameters n, p then it&#039;s probability mass function is given by   P  (  X  =  k  )  =                              (                            n        k                              )                          p    k    (  1  −  p      )          n      −      k       for all   0  ≤  k  ≤  n. The binomial coefficient                               (                            n        k                              )                     counts the possible ways to get k successful results in the total of the n trials.For example if   n  =  10 and   k  =  2, then you can obtain the two successes in trials 1 and 2, or in trials 1 and 3 or in trials 7 and 10 and so on. The binomial coefficient counts exactly all these possibilities and therefore it can&#039;t be omitted. It&#039;s value is equal to 1 when   k  =  1 or when   k  =  n - as you pointed out - and thus these are the only cases in which it can be omitted.

How can you tell when you need to use the binomial coefficient for probabilities?

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