Question

In a lottery, there are 2 000 000 tickets to be sold. The prizes are as follows below (picture attached):

a) Is this distribution uniform? Explain.

b) What is the random variable?

c) Show all possible outcomes for winning a prize in the lottery.

d) If a ticket cost $2 what is the expected profit per ticket?

Answer 1

@tcarroll010 @satellite73 where u guys at

Answer 2

@phi @jim_thompson5910 @timo86m @countonme123

Answer 3

I think each ticket is equally likely

Answer 4

The problem can also be found as an example problem #3:
http://www.dpcdsb.org/NR/rdonlyres/B4B42B52-D796-4298-9A7C-EAAFDE24F72D/77828/71ProbabilityDistributionsApr11.pdf

Answer 5

@phi what makes you think that. I thought the probability was different for winning each value prize.

Answer 6

only because some prizes can be won by more than 1 ticket

Answer 7

Prob of picking a particular ticket is 1/2000000

Answer 8

I know. For example, the probability of winning a 1000000 prize would be 1/66 correct? or would we say 1/2000000?

Answer 9

2,000,000 tickets are sold. the chance of getting the winner is 1/2,000,000

Answer 10

Here is the confusing part though. Even though it makes sense to say 1/2000000 as the probability of winning the highest prize, if we look at the second column, which is the sigma with x * P(x), that is the expected value column. Now, if we take the probability as 1/2000000, actually if we make the denominator of the probabilities 2000000 for each prize, the expected value comes to be really small. But the expected value should be the "expected" or average value of a prize. If we use 66 as the denominator, the expected value seems fine (I'm getting 19728 as expected value when I use 66 as denominator). I think in this table, we are to only consider the prizes as the denominator and not all the tickets. This is the main confusion here, 2000000 or 66 as denominator. Using 66 makes sense. What do you say? @phi

Answer 11

@phi where did you go

Answer 12

@tcarroll010 @phi @satellite73

Answer 13

First, you are doing example 3 under "Uniform Distribution"

Second, they don't say "consider only winning tickets"

I would say they want the expected value of a ticket (not the expected value of a ticket that won one of the prizes)

Answer 14

hmm lemme think

Answer 15

wait so in that sense, the expected value would be the average prize per ticket. In this case it would be 0.63125/ticket @phi

Answer 16

Is that what you mean? @phi

Answer 17

yes, that looks right.
if you fill in the table and add it up, that is what you get.

the final part, is expected "profit" = 0.63125 - 2

Answer 18

wouldn't that be 2-0.63125 since they get $2 for each ticket while they give 0.63125 for each ticket hence the profit per ticket is $1.36875? @phi

Answer 19

yes. It depends whose profit they are talking about. I was thinking of the buyer's profit, but it makes more sense to talk about the lottery's profit.

Answer 20

well it says "what is the expected profit per ticket", it would make sense to say that its referring to the profit that the lottery company makes.

Answer 21

yes, I agree

Answer 22

now, how is the distribution uniform? it's clear that the probabilities for each outcome is different. @phi so it would be nonuniform correct?

Answer 23

I already pointed out this is example 3 of a uniform distribution.... so we know it is uniform. (each ticket is equally likely, and if you plot the distribution, you get a box with length 2,0000,000 and height 1/2,000,000

Answer 24

this is like the dice example : equal chance of any particular # from 1 to 6
whether a particular outcome wins or loses does not determine the distribution.

Answer 25

@phi just because its under uniform distribution doesn't mean it is. I mean look at it. For dice example, the way roll each number is 1/6 so distribution is uniform. But here, the probability of winning a prize is different for each prize. then that means that it isn't a uniform distribution. Why do you say that is? I still don't make any sense of your reasoning.

Answer 26

c) Show all possible outcomes for winning a prize in the lottery.

This is the question I really don't know what to do for. @phi @satellite73

Answer 27

What do they mean by that