Theron has been playing Texas Hold \'Em for more than two years and has kept tra
ID: 3171022 • Letter: T
Question
Theron has been playing Texas Hold 'Em for more than two years and has kept track of his wins (and losses) to the nearest five dollars. He has learned from experinece that the probability he will lose $10 is 10%, lose $5 is 15%, win $0 is 25%, win $10 is 15%, win $25 is 30% and win $50 is 5%.
a) Construct a probability distribution table using Theron's experimental probabilities.
b) Based on this experimental data, on any given night what is Theron's expected value at the poker table?
c) Find the variance and standard deviation of his probability distribution.
Explanation / Answer
a) Construct a probability distribution table using Theron's experimental probabilities.
Solution:
The required probability distribution table is given as below:
Win ($) X
P(X)
-10
0.10
-5
0.15
0
0.25
10
0.15
25
0.30
50
0.05
Total
1.00
b) Based on this experimental data, on any given night what is Theron's expected value at the poker table?
Solution:
Here, we have to find the expected value for the poker table. The formula for expected value is given as below:
E(X) = XP(X)
The calculation table for expected value is given as below:
Win ($) X
P(X)
XP(X)
-10
0.10
-1
-5
0.15
-0.75
0
0.25
0
10
0.15
1.5
25
0.30
7.5
50
0.05
2.5
Total
1.00
9.75
E(X) = XP(X) = 9.75
c) Find the variance and standard deviation of his probability distribution.
Solution:
The formula for variance is given as below:
Variance = Var(X) = (X - mean)^2*P(X)
Standard deviation = SD = sqrt(variance) = sqrt[ (X - mean)^2*P(X)]
The calculation table for variance and standard deviation is given as below:
Win ($) X
P(X)
XP(X)
(X - mean)^2
(X - mean)^2*P(X)
-10
0.10
-1
390.0625
39.00625
-5
0.15
-0.75
217.5625
32.634375
0
0.25
0
95.0625
23.765625
10
0.15
1.5
0.0625
0.009375
25
0.30
7.5
232.5625
69.76875
50
0.05
2.5
1620.0625
81.003125
Total
1.00
9.75
246.1875
Variance = Var(X) = (X - mean)^2*P(X) = 246.1875
Standard deviation = sqrt(246.1875) = 15.69036328
Win ($) X
P(X)
-10
0.10
-5
0.15
0
0.25
10
0.15
25
0.30
50
0.05
Total
1.00
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