**The Binomial Distribution**

In (*p* + *q*)^{n}, the *p* and *q* represent the probabilities of alternative independent events, and the power n to which the binomial is raised represents the number of trials. The sum of the factors in the binomial must add to up to one; thus,

p+q= 1

Recall that when two independent events are occurring with the probabilities *p* and *q*, then the probability of their joint occurrence is *pq*. That is, the combined probability is the product of the independent events. When alternative possibilities exist for the satisfaction of the conditions of the problem, the probabilities are combined by addition.

EXAMPLE 2.25In two tosses of a coin, withp= heads = andq= tails =, there are four possibilities.

EXAMPLE 2.26Expanding the binomial (p+q)2 produces the same expression as in the previous example. Thus, (p+q)^{2}=p^{2}+ 2pq+q^{2}.

EXAMPLE 2.27When a coin is tossed three times, the probabilities for any combination of heads and/or tails can be found from (p+q)^{3}=p^{3}+ 3p^{2}q+ 3pq^{2}+q^{3}. Letp= probability of heads = 1/2 andq= probability of tails = 1/2.

The expansion of (*p* + *q*)^{3} is found by multiplying (*p*^{2} + 2*pq* + *q*^{2}) by (*p* + *q*). This process can be extended for higher powers, but obviously becomes increasingly laborious. A short method for expanding (*p* + *q*) to any power (*n*) may be performed by following these rules. (1) The coefficient of the first termis 1. The power of the first factor (*p*) is n, and that of (*q*) is 0. (Note: Any factor to the zero power is 1.) (2) Thereafter in each term, multiply the coefficient by the power of *p* and divide by the number of that term in the expansion. The result is the coefficient of the next term. (3) Also thereafter, the power of *p* will decrease by 1 and the power of *q* will increase by 1 in each term of the expansion. (4) The fully expanded binomial will have (*n* + 1) terms. The coefficients are symmetrical about the middle term(s) of the expansion.

**Single Terms of the Expansion**

The coefficients of the binomial expansion represent the number of ways in which the conditions of each term may be satisfied. The number of combinations (*C*) of *n* different things taken *k* at a time is expressed by

where *n*! (called "factorial *n*") = *n*(*n* – 1)(*n* – 2)… 1. (0! = 1 by definition.)

EXAMPLE 2.28Ifn= 4, thenn! = 4 (4 – 1) (4 – 2) (4 – 3) = 4 (3) (2) (1) = 24.

EXAMPLE 2.29The number of ways to obtain two heads in three tosses of a coin isThese three combinations are

HHT,HTH, andTHH.

Formula (2.1) can be used for calculating the coefficients in a binomial expansion,where ∑ means to sum what follows as

kincreases by one unit in each term of the expansion fromk= 0 ton. This method is obviously much more laborious than the short method presented previously. However, it does have utility in the calculation of one or a few specific terms of a large binomial expansion. To represent this formula in another way, we can letp= probability of the occurrence of one event (e.g., a success) andq= probability of the occurrence of the alternative event (e.g., a failure); then the probability that in n trials a success will occur s times and a failure will occurftimes is given by

**The Multinomial Distribution**

The binomial distribution may be generalized to accommodate any number of variables. If events *e*_{1}, *e*_{2}, …, e_{k} will occur with probabilities *p*_{1}, *p*_{2}, …, *p*_{k}, respectively, then the probability that *e*_{1},*e*_{2}, …, e_{k} will occur *k*_{1}, *k*_{2}, …, *k*_{n} times, respectively, is

where *k*_{1} + *k*_{2} + … + *k*_{n} = N.

Practice problems for these concepts can be found at:

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