Chapter 13.4, Problem 15PPS

a.

To determine

Calculate your team’s expected value for a hand.

Answer to Problem 15PPS

  $0.75$

Explanation of Solution

Given information:

A team can get $0$ points, $1$ point or $3$ points for a hand. The probability of your team getting $1$ point for hand is $60\%$ and $3$ points for a hand is $5\%$ .

Calculation:

A team can get $0$ points, $1$ point or $3$ points for a hand. The table describes the probability of each outcome.

  $\begin{array}{cc}\begin{array}{l}outcome\\ \left(po\mathrm{int}s\right)\end{array}& \mathrm{Pr}obability\\ 0& 1-\left(0.60+0.05\right)=0.35\\ 1& 0.60\\ 3& 0.05\\ total& 1\end{array}$

  $\begin{array}{l}Expectedvalue=\left(0\right)\left(0.35\right)+\left(1\right)\left(0.60\right)+\left(3\right)\left(0.05\right)\\ =0+0.6+0.15\\ =0.75\end{array}$

Hence, expected value for a hand is, $0.75$ .

b.

To determine

Design a simulation and estimate your team’s average value for a hand.

Answer to Problem 15PPS

  $0.84$

Explanation of Solution

Given information:

A team can get $0$ points, $1$ point or $3$ points for a hand. The probability of your team getting $1$ point for hand is $60\%$ and $3$ points for a hand is $5\%$ .

Calculation:

To simulate the card game, use a random number generator to generate integers. Use the integers $1through20$ . $1-7$ represent $0$ points, $8-19$ represents $1$ point, $20$ represent $3$ points. Run $50$ trials of the game and observed the frequency of outcomes it will vary between sets of $50$ trials. One such set of trials can be recorded in a frequency table,

  $\begin{array}{cc}\begin{array}{l}outcome\\ \left(po\mathrm{int}s\right)\end{array}& \mathrm{Pr}obability\\ 0& 14\\ 1& 33\\ 3& 3\\ total& 50\end{array}$

Divide each frequency with total frequency to get probability for each outcome.

  $\begin{array}{ccc}\begin{array}{l}outcome\\ \left(po\mathrm{int}s\right)\end{array}& Frequency& \mathrm{Pr}obability\\ 0& 14& 14/50=0.28\\ 1& 33& 33/50=0.66\\ 3& 3& 3/50=0.06\\ Total& 50& 50/50=1\end{array}$

  $\begin{array}{l}Averagevalue=\left(0\right)\left(0.28\right)+\left(1\right)\left(0.66\right)+\left(3\right)\left(0.06\right)\\ =0+0.66+0.18\\ =0.84\end{array}$

Hence, average value for a hand is, $0.84$ .

c.

To determine

Compare expected and average value for a hand.

Answer to Problem 15PPS

  $AlmostEqual$

Explanation of Solution

Given information:

A team can get $0$ points, $1$ point or $3$ points for a hand. The probability of your team getting $1$ point for hand is $60\%$ and $3$ points for a hand is $5\%$ .

Calculation:

Expected value is $0.75$ and average value is $0.84$ both the values are $AlmostEqual$ to each other.