Question

trenton plays a game with a biased coin with $p(heads)=0.63$ and $p(tails)=0.37$. he plays each game by tossing the coin once. if he tosses a head, he pays $8. if he tosses a tail, he wins $13. complete the probability distribution table. probability distribution table $x$ $p(x)$ 13 what is your long - term expected profit per game? $ what is the standard deviation? $ question help: message instructor post to forum

Explanation:

Step1: Complete the probability - distribution table

When $x=- 8$ (losing $8$ when getting heads), $P(x)=0.63$. When $x = 13$ (winning $13$ when getting tails), $P(x)=0.37$.

Step2: Calculate the expected value $E(X)$

The formula for the expected value of a discrete - random variable is $E(X)=\sum_{i}x_{i}P(x_{i})$. So, $E(X)=(-8)\times0.63 + 13\times0.37=-5.04+4.81=-0.23$.

Step3: Calculate $E(X^{2})$

$E(X^{2})=(-8)^{2}\times0.63 + 13^{2}\times0.37=64\times0.63+169\times0.37 = 40.32+62.53 = 102.85$.

Step4: Calculate the variance $Var(X)$

The formula for the variance is $Var(X)=E(X^{2})-[E(X)]^{2}$. Substitute $E(X)=-0.23$ and $E(X^{2}) = 102.85$ into the formula: $Var(X)=102.85-(-0.23)^{2}=102.85 - 0.0529=102.7971$.

Step5: Calculate the standard deviation $\sigma$

The standard deviation $\sigma=\sqrt{Var(X)}$. So, $\sigma=\sqrt{102.7971}\approx10.14$.

Answer:

Probability - distribution table:

$x$	$P(x)$
13	0.37

Expected profit per game: -$0.23$
Standard deviation: $\$10.14$