PROBLEM SOLVING PACKET #29



APPL IN SEQ MATH NAME______________________

The Math of Gaming

The following activity in NO way promotes or endorses gaming. Its purpose is to show the relatively low probability of winning and its impact on a community.

Gov. Cuomo has proposed gaming in New York State, outside of Native American casinos. Supporters of casinos believe it will bring a much-needed economic boost to the area. Opponents believe casinos bring about increased levels of crime to an area.

Let us examine the probabilities of “winning” games of chance. Specifically, what is the probability of winning the New York State Lotto and the probability of being dealt a straight (5 cards in numerical order) in poker?

LOTTO What are the odds of winning a prize?

The difference between probability and odds…

The PROBABILITY of an event occurring is expressed as a fraction,

Number of ways an event can happen/Total number of possible outcomes e.g. Prob of rain this week is 2/7 or 29%, 2 out of 7 days

The ODDS tell you how many times an event will happen and how many times it won’t happen

Chances for:Chances against e.g. Odds of rain this week are 2:5, 2 days rain, 5 days dry

The object of the game is to choose 6 numbers from 1 to 59

Let’s look at the odds of winning. You have to pick ALL 6 numbers correctly. The answer is dependent upon whether or not order is important and whether or not the numbers are “replaced” after they are drawn. The Lotto game is “without replacement” since a number cannot be drawn twice. That is, once a number has been drawn, it is not placed back into the pool of eligible numbers.

In lottery games, the order in which the numbers are chosen is unimportant. All that matters is that you have the same numbers on your card as were drawn by the lottery officials.

Since order does not matter, we are going to use a COMBINATION. If order mattered, we would use a permutation.

In general, the number of ways to choose r objects n ways is n C r = n!

r! (n-r)! n= 59 numbers from which

to choose from

r= 6 numbers are to be

drawn

59 C 6 = 59(

6((53() = [pic]

The odds of winning the big prize is _________________

STRAIGHT IN A HAND OF FIVE CARD STUD POKER

The probability of getting a straight (5 cards in numerical order)

There are 52 cards in a deck.

1st card dealt could be anything from 2 to 10 (if we start above 10, there will not be enough cards to make a straight –jack, queen, king, and ace)

From 2 to 10 there are 36 cards

Prob (1st card) = 36 /52

Prob (2nd card one higher) = only 4 cards qualify (one in each suit) out of 51 = 4 /51

Prob (3rd card one higher)= 4 /50

Prob (4th card one higher) = 4 /49

Prob (5th card one higher) = 4/48

Number of ways (combinations) of dealing same 5 cards in ANY order = 5(

So, Prob (getting a straight) = [pic]__________________

The probability of drawing a straight is less than 1%

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