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A probability distribution in general and a binomial example.

A probability distribution in general and a binomial example. X P(X) Cumulative Probability 0 P(0) = a P( 0 or less) = a 1 P(1) = b P(1 or less) = a + b 2 P(2) = c P(2 or less) = a + b + c 3 P(3) = d P(3 or less) = a + b + c + d 4 P(4) = e P( 4 or less) = a + b + c + d + e

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A probability distribution in general and a binomial example.

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  1. A probability distribution in general and a binomial example.

  2. X P(X) Cumulative Probability 0 P(0) = a P( 0 or less) = a 1 P(1) = b P(1 or less) = a + b 2 P(2) = c P(2 or less) = a + b + c 3 P(3) = d P(3 or less) = a + b + c + d 4 P(4) = e P( 4 or less) = a + b + c + d + e 5 P(5) = f P( 5 or less) = a + b + c + d + e + f Say some dude works away from his house Monday through Friday. He packs a lunch and always buys some sort of drink from a machine during lunch. His choices are either water or one of several very delicious soda pops. For now let’s say he closes his eyes and picks the drink. Let’s let X be the number of days in the 5 day work week that he gets a soda. For now say the probabilities are assigned. It must be that in the P(X) column each letter I have is a number either 0 or 1 or between 0 and 1 and it must be that the sum of the letters adds up to 1.

  3. What is the probability that during the week he has 2 soda pops? We could write this as P(2). Well the P(X) in the row with a 2 is the answer and we have the value c. What is the P(4)? P(4) = e. What is the probability that X is three or less? This might be written P(X <=3). The answer is seen in the table as a + b + c + d. What is probability that X is less than 3? In the context of a discrete variable less than 3 is the same as 2 or less and so the answer would be a + b + c. What is the probability X is more than 3? More than 3 is really 4 or 5 here, but we can also take 1 – P(X <= 3) = 1 – (a + b + c + d) and since 1 = a + b + c + d + e + f, P(X > 3) = a + b + c + d + e + f – (a + b + c + d) = e + f.

  4. X P(X) Cumulative Probability 0 P(0) = a P( 0 or less) = a 1 P(1) = b P(1 or less) = a + b 2 P(2) = c P(2 or less) = a + b + c 3 P(3) = d P(3 or less) = a + b + c + d 4 P(4) = e P( 4 or less) = a + b + c + d + e 5 P(5) = f P( 5 or less) = a + b + c + d + e + f In this area our book uses a column labeled k instead of X, does not give the second column, gives the third column without the last row. Now, the last row of the third column does have the value 1. So, that is easy enough to remember. So, in this context our book has k Cumulative Probability 0 P( 0 or less) = a 1 P(1 or less) = a + b 2 P(2 or less) = a + b + c 3 P(3 or less) = a + b + c + d 4 P( 4 or less) = a + b + c + d + e The book says that in the cumulative probability column in each row we have P(X ≤ k). This means, for example, that in the k = 2 row, the P(X ≤ k) = P(X ≤ 2) = a + b + c.

  5. So, our book gives us the cumulative probability column. Sometimes we want to have the P(X) column values that I started with. How do we get each P(X) value from what we have? X P(X) Cumulative Probability 0 a P( 0 or less) = a 1 a + b – a = b P(1 or less) = a + b 2 a + b + c – (a + b) = c P(2 or less) = a + b + c 3 a + b + c + d – (a + b + c) = d P(3 or less) = a + b + c + d 4 a + b + c + d + e – (a + b + c + d) = e P( 4 or less) = a + b + c + d + e 5 1 – (a + b + c + d + e) = f our book doesn’t have this amount – you know it = 1 So, in any row to get a P(X) take that row’s cumulative probability and subtract of the previous row’s cumulative probability. Of course in the X = O the P(0) = the cumulative probability. The book doesn’t include the X = 5 row but to get P(5) you take 1 minus the previous row’s cumulative probability.

  6. Now, let’s return to our story of the dude and his saga of eating lunch at work. Say he has made a special box. It kind of looks like a box of Tic Tac Mints. Are you familiar with these mints? They are very bad for you. If you ever get some you should give them all to me. Wow, this is like a whole rack of tic tacs! Totally awesome! Well, the dude has only one box and it is made of wood. Inside the box he has ten little metal slugs. Each slug is round and all ten are the exact same size. The box is big enough that he can shake them up. Some days he takes out the box and jiggles the box just to hear the slugs bouncing around the walls of the box. Now, You should know 3 of the slugs are red and seven are ocean blue. When it becomes lunch time the dude gets his box and gives it a shake. Then he turns the box on its side and opens the lid. But he can not see inside the box. And he jiggles the box until a slug falls out of the box. If the slug is red he buys a pop and if the slug is blue he buys a water. The dude knows the odds are against him getting a pop and he wants it that way because he is getting older and knows pop can pack on the pounds and he wants to conserve. But, at the same time he wants life to be worth living!

  7. Let’s call the x the number of days in the 5 day work week the dude gets a pop. So you can see x could be anywhere from 0 up to 5. In the binomial table we see when n = 5 and p = .3 (on any day the prob of a pop = .3) k 0.30 0 .1681 1 .5282 2 .8369 3 .9692 4 .9976 I want you to be able to then translate this to X P(X) Cumulative Probability 0 .1681 .1681 1 .5282 - .1681 .5282 2 .8369 - .5282 .8369 3 .9692 - .8369 .9692 4 .9976 - .9692 .9976 5 1 - .9976 1 What is the probability that the dude gets a pop every day during the work week? 1 - .9976 = .0024, a very low chance, indeed!

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