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QUICKHELP MAT109 Probability & Statistics The binomial probability applies to experiments for which the following is true: VOCABULARY & NOTATION binomial: comprised of only two categories ex. binomial number system has only 1's and 0's n # of trials x # of successes f p= f Binomial Probability a) the experiment consists of repeated trials b) each trial is independent c) each trial has only two possible outcomes p p = probability of success in one trial; p x = probability of x successes occurring Situations in the real world which have binomial probabilities are flipping a coin (head or tail), hitting a bulls-eye (hit or miss), making a foul shot (yes or no), and birth of a child (b l) q = probability of failure in one trial; n x The binomial probability function is a handy formula to calculate probabilities in binomial situations. n P (x) = f p p x qn - x x Let's break this down: P (number of successes) = f Example 1 n = # trials x = # successes p f prob of success # of success p prob of f p failure # of failures A balanced coin is tossed 4 times. What is the probability of getting 3 heads? n = 4 tosses x = successes, i.e., heads = 3 p = 2 , probability of getting a head in one toss 4 13 1 4- 3 P (3heads in 4 tosses) = f p f d1 - n p 2 3 2 4 13 1 1 = f pf pd 2 n 3 2 = _4 i _0.125i _0.5i = 0.25or 25% Example 2 5 cards are drawn from a deck, one at a time and then replaced. What is the probability that at least 3 of them will be red? First, let's determine if this is a binomial situation, using a) through c) from above. a) repeated trials? b) independent trials? Yes. 1 card at a time, 5 times Yes. Replacing the cards after they are drawn keeps each trial unaffected by previous outcomes. 3 2 5 c) only two outcomes? Yes. Red or not-redp _(black). 3red in5 draws i = f p d 1 n d 1 n = 0.3125 2 3 2 The total 5 1 4 1 1 It is binomial, so we can use the formula. probability p _4red in5 draws i = f pd n d n = 0.15625 2 4 2 is the sum 5 1 5 1 0 ** The probability of each situation which is a success must be of these, or p _5red in 5drawsi = f pd n d n = 0.03125 calculated separately then added together. MAT109 Probability & Statistics Math Center Miller Library, x7862 5 2 2 Binomial Probability http://mathcenter.washcoll.edu