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Binomial distribution And general probability distributions... Random variables • A number is assigned to an outcome that is determined by chance. • Definition denoted by uppercase letters at the end of alphabet, such as X, Y, Z • Possible values denoted by corresponding lowercase letters x, y, z Examples: random variables • X = number of beers randomly selected student drank last night, x = 0, 1, 2, … • Y = weight of a randomly selected Big Mac, 0.20 x 0.30 pound • Y = time spent daily on phone of randomly selected student, 0 x 1440 minutes Discrete probability distribution All possible outcomes of the r.v. and the probability that each outcome will occur. Example Toss a fair die. Let X = value on die. Then P(X=x) = 1/6, x =1, 2, 3, …, 6 Continuous probability distribution All possible outcomes of the r.v. and a formula that allows us to determine the probability of a range of values. Example Let X = blood alcohol level of randomly selected student. Use formula to find P(X 0.10) Example: binomial r.v. • Pick one number between 0 and 9. • Let X = number of students picking certain number. • If random picking, then probability random student picks special number is 0.10. Binomial r.v. A special kind of discrete r.v having the following characteristics: • n identical “trials”: n students pick # between 0 and 9 • 2 possible outcomes denoted “success” or “failure”: student picks special # or not • independent trials: what one student picks doesn’t affect what another student picks • p = P(“success”) is same for each trial: if random picking, each student has probability of 0.10 of picking special # Is X binomial? Probability student smokes pot regularly is 0.25. College administrator surveys students until finds one who smokes pot. Let X = number of students surveyed Is X binomial? Unknown to quality control inspector, crate of 50 light bulbs contain 3 defective bulbs. QC inspector randomly selects 5 bulbs (“without replacement”). Let X = number of defective bulbs in inspector’s sample. Is X binomial? Unknown to us, the probability an American thinks House Judiciary Committee is mishandling the “Clinton thing” is 0.78. Gallup poll surveys 960 Americans. Let X = number of Americans in sample who think HJC is mishandling the Clinton thing. Recall example • Pick one number between 0 and 9. • Let X = number of students picking certain number. • If random picking, then probability random student picks special number is 0.10. Example: binomial r.v. Let 3 students pick. Y = special #, N = not special # Outcome X P(X=x) NNN 0 1(0.1)0(0.9)3 YNN NYN NNY YYN YNY NYY YYY 1 3(0.1)1(0.9)2 2 3(0.1)2(0.9)1 3 1(0.1)3(0.9)0 Binomial probability distribution P(X = x) = (# of ways x occurs) × px × (1-p)n-x = n!/[x!(n-x)!] × px × (1-p)n-x Where “n-factorial” is defined as n!= n (n-1) (n-2) … 1 and 0! = 1 Examples: n! 5! = 5 × 4 × 3 × 2 × 1 = 120 4! = 4 × 3 × 2 × 1 = 24 3! = 3 × 2 × 1 = 6 2! = 2 × 1 = 2 1! = 1 Example: binomial formula P(X = x) = n!/[x!(n-x)!] × px × (1-p)n-x Assume n = 3 and p = 0.2. Then: P(X = 0) = 3!/[0!(3-0)!] × 0.20 × (0.8)3-0 = 6/(1×6) × 1 × 0.83 = 0.512 P(X = 1) = 3!/[1!(3-1)!] × 0.21 × (0.8)3-1 = 3 × 0.2 × 0.82 = 0.384 Example (continued) P(X = 2) = 3!/[2!(3-2)!] × 0.22 × (0.8)3-2 = 3 × 0.04 × 0.8 = 0.096 P(X = 3) = 3!/[3!(3-3)!] × 0.23 × (0.8)3-3 = 1 × 0.008 × 1 = 0.008 Note: 0.512 + 0.384 + 0.096 + 0.008 = 1 Using binomial probabilities to draw a conclusion If students do indeed pick a number between 0 and 9 randomly, how likely is it that we would observe the sample we did? Can we conclude that students do not pick numbers randomly? Using binomial probabilities to draw a conclusion Could the space shuttle Challenger disaster of January 28, 1986 have been better predicted? And therefore prevented? Moral • Probability calculations are used daily to draw conclusions and make important decisions. • Calculated probabilities are accurate only if the assumptions made are indeed correct. • Always check to see if your assumptions are reasonable.