Download MW.wksht15.04 p.603-607 expected value

Statistical Reasoning in Sports Worksheet#15.4 (p.603-607)
Name _______________________ Per._____ Date _____________
Probability distributions and expected value
In a sports setting, often there is a variety of possible results, each with its own probability. The individual
probabilities add to 100%, but some are more likely to occur than others. Here is an example: at Indian Springs
golf course, the 1st hole is a 350 yard long par 4. This table shows the scores that golfers made on this hole in a
recent season. In other words, during that season, 635 golfers had a score of 4 on that hole, 982 golfers had a
score of 5, etc.
Score
2
3
4
5
6
7
8
Totals
number
of
golfers
3
148
635
982
378
49
30
2225
Calculate the percentage of times each score was made, leaving the answer in decimal format correct to 4 decimal
places. The hole was played 2225 times, so the percentage of golfers who scored 4 is 635 out of 2225, or 635 
2225, and so on for each of the other scores. Put each answer in the table below: (note: answer correct to 4
places because these answers will be used in a later calculation)
Score
2
3
4
5
6
7
8
decimal
Totals
1.0000
If we replace “Score” with “x” and “decimal” with “P(x)”, the table is now a probability distribution. It shows the
probability of getting each of those scores for a randomly selected golfer. Then we can create an overall
expected value, which is the expected score on this hole. Multiply each score times the percentage of times that
score was made, leaving the answer in decimal format correct to 4 decimal places, and add up all the results.
x
2
3
4
5
6
7
P(x)
8
Totals
1.0000
Score
times %
E(x): Expected value, or expected score on this hole ------
You would interpret the expected value as follows:
If I were to randomly select golfers over and over who played the
Indian Springs golf course hole #1, their average score would be about
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Practice problem 1:
Let x = the number of points scored when a high school football team starts on their own 20 yard line. The
probability distribution of x is shown in the table below.
( a ) Calculate the expected value of x
( b ) Interpret the expected value that you just calculated.
x
0
3
7
P(x)
0.62
0.13
0.25
Totals
Expected values when a football team goes for
1 or 2 extra points after a touchdown.
Background: When a team scores a touchdown they get 6 points and they also get to try for 1 or 2 extra points.
The ball is placed at the 3-yard line. They get 1 point for a kick through the goal posts, or they get 2 points for any
other play that moves the ball across the goal line. Most teams would have a 95% probability of success to kick for
the 1 extra point but only a 50% probability of success to gain the yards and score 2 points.
Potential reward: 2 points is more than 1 point !
Potential risk: If the play doesn’t work then the team gets 0 points.
GO FOR 1:
GO FOR 2:
Use a probability distribution to calculate the expected
value (and therefore the expected number of points
scored) when a team tries for 1 point after a touchdown:
x
0
1
P(x)
Use a probability distribution to calculate the expected
value (and therefore the expected number of points
scored) when a team tries for 2 points after a
touchdown:
Totals
x
1.0000
P(x)
Score
times %
0
2
Totals
1.0000
Score
times %
E(x): Expected value, or expected score 
E(x): Expected value, or expected score 
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Practice problem 2:
Let x = the number of runs scored when a major league baseball team has a runner on 3rd base with 2 outs (there
may also be other runners on base). The probability distribution of x is shown in the table below (total runs scored
in the inning from that point forward).
( a ) Calculate the expected value of x
( b ) Interpret the expected value that you just calculated.
x
0
1
2
3
P(x)
0.68
0.22
0.08
0.02
Totals