#### Solutions to 99/08/23 Problems

Use ProbSim to investigate these situations. Use the blank screens as worksheets as you think through all the decisions and assumptions you need to make in order to have ProbSim simulate the situation accurately.

### Problem 1.

In all poker-like card games, the "most unusual" hand is the winner (i.e., the hand least likely to occur over the long run).

Horace, Hillary and friends were playing cards -- "guts". Guts is a game where people are dealt 3 cards each, and when the dealer says "1, 2, 3, drop." People thinking their hands aren't good enough to win drop. All the people thinking they have a good enough hand will keep their cards.

Horace and Hillary remembered that, in 5-card poker, a flush (all cards the same suit) beats three of a kind (three cards the same face value). But they didn't know if that should be the case in 3-card poker.

Use ProbSim to resolve this question.

#### Solution to Problem 1

The question is asking us which hand occurs less frequently over the long run. The way to use ProbSim to resolve this question is to set it up to deal three cards. We will have to run ProbSim twice -- once to investigate the relative frequency of 3 of a kind and once to investigate the relative frequency of 3 of the same suit. I say "relative frequency" instead of just "frequency" because we are interested in the fraction of the time that we get each hand when 3 cards are dealt at random from a standard deck of cards.

Assumptions:

1. We are sampling from a standard card deck: 52 cards in a deck -- 4 each of Ace, 2, 3, ..., 10, J, K, Q. Also, there are 13 cards in each of 4 suits -- Hearts, Clubs, Diamonds, and Spades.
2. We are sampling one hand (a set of three cards) at random. That is, it is impossible to predict which card will be dealt any time a card is dealt.
3. We are mixing the cards well prior to each sample. This is to ensure that we do not bias the results of the next sample because of the way we replace the current sample. Mixing the cards assures us that, over the long run, we do not bias the results.

#### Part a -- 3 of a kind

First we need to set up the mixer so that we are sampling what we have in mind. Since we are paying attention to the face values of cards and not their suits when we look for 3-of-a-kind, we want to fill the mixer with "face values." We could type "A", "2", "3", ...,"10", "J","Q","K", but that isn't necessary. All we need are 13 different labels, and 4 of each label. I've used the RANGE feature of ProbSim (at the right) to do this.

After setting up the mixer using the RANGE feature, ProbSim's screen looks as below.

I've set up ProbSim to select 3 "cards" from the mixer, without replacing a card once picked (but putting all the cards back after taking one sample). It will repeat this 1000 times when I click Run. (Notice that I clicked "F" to make ProbSim run Fast. Otherwise it would actually show each card being selected, and that would take cause the program to take long time to collect 1000 samples.)

Once I ran ProbSim and got some data I needed to decide how to analyze the data.

We are interested in 3 cards of the same kind. So, (1,1,1), (2,2,2), ..., and (13, 13, 13) all would give us 3 of a kind. We could count each one separately, but that would mean that we would have to do 13 separate analyses. We can say "three of a kind" in general by using variables. (V1,V1,V1) means "count any instance that has any one number appearing three times".

These are the results of counting the number of 3-of-a-kind out of 1000 hands sampled from the deck (remember, we assume each hand is placed back into the deck and the deck is well shuffled).

##### Results of 3-of-a-kind analysis

It appears that we will get 3-of-a-kind about 1/2 of 1% of the time. That is, it seems that we will get 3-of-a-kind in less than 5 out of every 1000 3-card hands.

#### Part b -- 3 of the same suit

We have the same assumptions, but a slightly different set up for the mixer. We will have 52 cards again, but the only thing we care about is their suit. So, it is as if we have 4 labels (one each for Hearts, Diamonds, Clubs, and Spades) and 13 of each label.

To analyze the data generated by drawing samples from this population we need to direct the program to count those instances where all three cards are of the same suit. To do that, we tell ProbSim to count only those elements that have three cards of the same label, just as before.

I repeated the repeat the process of gathering 1000 different samples four times. The analyses of these four repetitions of gathering 1000 samples are shown below.

The results suggest that when we take 3 cards at random from a standard deck a large number of times, about 5% of the time we will get three cards of the same suit.

Conclusion

It seems that when we take 3 cards at random from a standard deck a large number of times, it will be far more unusual to get 3 of a kind than it will be to get 3 cards of the same suit. Therefore, a hand of 3-of-a-kind should beat a 3-card flush.

### Problem 2.

The Gallup company surveyed 20 past graduates of Metro Tech, asking them to if they were satisfied with the education that Metro gave them. Only 61% of the graduates said they were very satisfied. However, the administration claims that over 80% of all past graduates are very satisfied

How unusual would a result like this be if, as MT administration claims, 80% of all graduates are very satisfied with their Metro education?

#### `Solution`

It is imperative the you have an appropriate image of the underlying situation. First, there is a (presumably large) population of Metro Tech graduates out there, and a certain percent of them are satisfied. We don't know for sure what that percent is. The administration claims that 80% of them are satisfied, and the question asks us to suppose (for the time being) that the administration is correct.

The Gallup company selected 20 people at random from the population of MT graduates. The sample they selected had a satisfaction rate that is very different from what the administration claims is true for the whole population. So, we are being asked to ASSUME that 80% of some large number of people are satisfied, and INVESTIGATE how unusual it would be to get a sample of 20 people, selected at random, that has around 60% of it being satisfied. That is, we are asked to investigate what fraction of the time we get at most 12 of 20 people being satisfied when we select the 20 at random from a population of which 80% is satisfied.

We are not given the actual number of graduates, so we just make up a large number, splitting them so that "Satisfied" makes 80% of the total. This is our population from which samples of 20 "people" (actually, labels) will be selected.

We need to set up ProbSim to take samples of size 20, without replacement (we must not survey the same person twice).

The administration's claim is in question, so we are interested in the fraction of samples from this population that have at most 60% saying they are satisfied. Another way to put this is that we can examine what fraction of the time we get at least 8 unsatisfied graduates when we select 20 graduates at random. This will tell us how likely it is that we get a sample that, like ours, will throw the administration's claim into question.

The anlaysis below shows that we counted those instances where we got at least 8 of 20 saying "unsatisfied". Repeating this process -- taking 1500 samples and counting the number that have at least 8 of 20 saying "unsatisfied" -- we see that, when we assume that the population is split 80/20 as the adminstration claims, around 3 percent of the time we would get samples having at least 8 saying "unsatisfied".

#### Conclusion

It seems that when we draw many samples of size 20 at random from a population that is actually 80% "Satisfied", about 3% of those samples will have 60% or less saying "Satisfied". That is, if the administration is correct, then samples like the one Gallup took are very unusual.

So, there seems to be two possibilities. (1) Gallup did a lousy job sampling from this population -- getting a sample that is far from representative, or (2) the administration is inflating its figures for political purposes and the actual percent of satisfied graduates is much lower than 80%.

### Problem 3.

The Gallup company asked 21 adults, selected at random, whether they attended church or synagogue during the past week. Fifty percent said they had. The Harris company performed an identical survey. In their survey, only 33% responded "yes".

Use ProbSim to investigate whether the two polls contradict one another.

#### Solution

First, "an identical survey" means that they asked the same questions of another randomly selected group of 21 people. Harris did not ask the same 21 people that Gallup asked.

Here's a quick analysis:

One way to approach this question would be to say, "Suppose Gallup's poll accurately reflects the population of church goers. How unusual , then, would be samples like Harris'?" or "Suppose Harris's poll accurately reflects the larger population. How unusual, then, would be samples like Gallup's?"

The only way that these two polls would not be contradictory is if the actual population from which the samples were drawn has some percentage between 50 and 33 that would result in samples like Harris' and Gallup's both being more common than "unusual".

#### Results

The above table shows a "Count All (Unordered)" analysis of 1500 samples (each of size 21 taken from a population that has 42% church goers).

Samples like the Harris' (33%), which would have 7 or fewer " Go's", happen about 28% of the time when sampling from a population having 42% "Go's". Samples like Gallup's, which would have 11 or more "Go's" out of 21, happen about 23% of the time when sampling from a population having 42% "Go's".

#### Conclusion

So, it seems not unusual to collect two polls by selecting 21 people at random from a population that has 42% of it attending church in the past week and have one poll getting 33% attending church and the other getting 50% attending church. They both could in fact be good polls (no bias in their selection procedures).