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Applications of Probability

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Objective

In this concept you will learn about some of the more entertaining applications of probability and statistics.

Concept

Do football players or coaches need to understand and use statistics? What about track and field competitors or swimmers? Can an understanding of probability make you more likely to win games of chance?

Read on to see how becoming an expert statistician can help you be more successful in ways you might never have imagined.

Guidance

Competitive sports are extremely popular in countries all over the world. In some countries, players can make huge salaries in competitive sports, and even countries where professional sports are outlawed encourage players to compete for national fame in the Olympics.

The obvious part of becoming a successful athlete is the vigorous physical training. Naturally a player will want to develop the instincts and physical prowess appropriate to his or her chosen sport. The less obvious, but no less important, part of becoming a champion athlete is developing and applying an understanding of the statistically important research about the sport.

Baseball pitchers need to know the particular types of pitches to throw at each hitter on an opposing team. By reviewing each hitter’s past records of strikes, fouls, base hits, and home runs based on each type of pitch, the pitcher will know who is more susceptible to fast balls, curves, etc., and will be a much more valuable member of his or her team.

Football players are equally dependent on statistics. A quarterback who has no idea which teammates are most likely to catch a particular throw based on previous performance would not have much chance of leading his or her team to victory. Coaches without a detailed understanding of what training methods are statistically most successful for developing the particular skills a team needs are not liable to lead their teams to victory.

Another pastime with eager participants all over the world is playing games of chance. Whether involving cards, dice, marbles, or anything else, any game of chance involves some level of probability by definition.

By becoming an expert on the likelihood of a particular card showing up in a random shuffle, or a specific roll turning up on a fair die , a clever player can significantly improve his or her chances of winning more often. Additionally, by studying and applying the statistical data of his competitors, a skilled player will know when to bet high on a bluff, or even hold an otherwise good play for a better time.

Example A

The Jets, a nationally competitive football team, are planning to play the #1 ranked team in their league in 2 weeks. What kind of statistics might they want to pay particular attention to as they research their future opponents?

Solution:  Very generally, the Jets might want to know whether their opponents are more offensive or more defensive, whether they tend to ‘run the ball’ or pass more often. It would also be beneficial to identify the most dangerous players on the other team so that they can be specifically focused on. All of these things could be relatively easily calculated by a statistician reviewing past games, and would represent a significant advantage to the Jets as they go into the competition.

Example B

Xio is playing a card-based game of chance with two other people. The game is played in rounds. In each round each player is dealt one card from the deck, and each player bets that his or her cards have the highest point value in that round.

If the deck starts with 4 each of the cards numbered 1-10, and Xio knows that there have been two 9's, and three 10's out of the 15 cards already played, how confident should he be that the 9 he was just dealt will represent a positive outcome the next round?

Solution:  There are 24 unknown cards remaining in the deck: 40(original deck) – 15(dealt previously) – 1(Xio’s card), and only a 10 will beat Xio’s 9. If there is only a single 10 remaining out of the 24 unknown cards, Xio can be pretty confident that his 9 will at least tie in this round.

The actual chance is  $\frac{1}{24}$  or approximately 4% for the first player, and  $\frac{1}{23}$  or also approximately 4% for the second:

$4 \% + 4 \% = 8 \%$  chance that Xio will lose this round, pretty good odds

Example C

Jane is playing a game of chance with dice. The object is to roll two dice as many times in a row as possible  without  rolling a 7. If Jane has already rolled 5 times without getting a 7, how likely is it that she will roll a 7 on the next toss?

Solution:  The first, and perhaps somewhat  counter-intuitive , point to note is that dice do not have any memory! That means that Jane’s chances of rolling a non-seven on her next roll are exactly the same as they were on her first roll, so we need to get an idea of how likely a roll of seven is on any given toss.

Seven is a common key number when discussing two “fair” (each number has the same chance of appearing) six-sided dice. This is because there are more ways to roll a seven than any other combination. The chart below shows the number of ways to roll each possible value of 2-12:

 2 $1+1$ 3 $1+2$ $2+1$ 4 $1+3$ $3+1$ $2+2$ 5 $1+4$ $4+1$ $2+3$ $3+2$ 6 $1+5$ $5+1$ $2+4$ $4+2$ $3+3$ 7 $1+6$ $6+1$ $2+5$ $5+2$ $3+4$ $4+3$ 8 $2+6$ $6+2$ $3+5$ $5+3$ $4+4$ 9 $3+6$ $6+3$ $5+4$ $4+5$ 10 $4+6$ $6+4$ $5+5$ 11 $5+6$ $6+5$ 12 $6+6$

We can see by counting on the chart that there are thirty-six possible combinations of two six-sided dice. There are six different combinations that result in a 7, thus, the chances of rolling a 7 would be:

$\frac{6}{36}=\frac{1}{6} \ or \ 17 \% \ chance \ of \ rolling \ a \ 7$

It is clear that at the beginning of each roll, Jane’s chances of not getting the infamous 7 are really quite good. In later lessons we will discuss the overall probability of rolling a 7 six times in a row, which is quite different than the probability of rolling a 7 on a sixth toss.

Concept Problem Revisited

Do football players or coaches need to understand and use statistics? What about track and field competitors or swimmers? Can an understanding of probability make you more likely to win games of chance?

We can certainly see, based on the examples above, that statistics and probability calculations are integral to sports of all kinds, and that understanding the probabilities involved with any given outcome can help you make smart decisions in games of chance.

Vocabulary

fair die  is a die with an equal chance of landing on any side. An  unfair die  would be more likely to land on a particular number than the others.

sample space  is the set of all the possible outcomes of an event.

sample point  is just one of the possible outcomes

If something is  counterintuitive , it is not what you might initially guess. For instance, it seems counterintuitive that a feather in a vacuum will drop like a stone, but it is a fact regardless.

An  event  is a particular occurrence in a series: one roll of a die, one flip of a card, etc.

Guided Practice

1. Suppose you have a single six-sided die, what is the probability of rolling an odd number for a game of chance?
2. Suppose you are playing a game where you flip two coins and try to guess how they will land. Would it be better to guess that both land with the same side up or that they would land with different sides showing?
3. If you draw a card at random from a single deck, what is the sample space of possible outcomes?

Solutions:

1. The sample space would be each of the possible numbers: 1, 2, 3, 4, 5, and 6. Since you can only roll one of them, and all are equally likely, the probability of any particular number is  $\frac{1}{6}$ .
2. If you flip two coins, the sample space is: {(H, T), (H, H), (T, T), (T, H)}. Since there are four possibilities, and two show the same side, while two show different sides, the probabilities are equal: 50%.
3. The sample space is the entire deck. Ace, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King of each of the four suits (The probability of pulling any single particular card is  $\frac{1}{52}$ ).

Practice

1. Identify four kinds ofentertainment or sports that benefit from statistics.
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2.
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2. How could it be worthwhile for a pitcher to study the type(s) of pitch that a specific hitter is most likely to score a run on?
3. How could a strategy gamer benefit from studying the play style statistics of an opposing team?
4. How do players of online multiplayer games like World of Warcraft or Second Life use statistics?
5. Is there a benefit to learning statistics before playing games of chance? Give an example.