A gambler plays a game of chance in which he has 60% chance of winning. If he wins, he collects N1,800 but if he loses he pays N2,300.
Required:
If he loses alternate games,
i. Obtain the sample spaces respectively for 12 games and 14 games
(ii. Calculate his expected returns for 12 games and 14 games.
iii. Advise the gambler if he initially staked N1000?
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
However, I will give you a start.
Unless you have a typo, he will lose 2,300-1,800 = 500 every two games. Personally, I would advise him not to gamble.
To solve this problem, we need to break it down into steps:
i. Obtaining the sample spaces for 12 games and 14 games:
The sample space refers to all the possible outcomes of the events. In this case, the gambler can either win or lose each game.
For 12 games:
Since the gambler loses alternate games, there will be 6 wins and 6 losses in 12 games. We can create the sample space as a string of "W" for win and "L" for loss:
Sample space for 12 games: WLWLWLWLWLWL
For 14 games:
Similarly, since the gambler loses alternate games, there will be 7 wins and 7 losses in 14 games.
Sample space for 14 games: WLWLWLWLWLWLWL
ii. Calculating the expected returns for 12 games and 14 games:
The expected return is the average amount the gambler can expect to win or lose in each game, considering the probabilities of winning and the amount won or lost.
Expected return = (Probability of winning * Amount won) - (Probability of losing * Amount lost)
For 12 games:
Probability of winning = 60% = 0.6
Amount won = N1,800
Probability of losing = 40% = 0.4
Amount lost = N2,300
Expected return for 12 games = (0.6 * 1,800) - (0.4 * 2,300)
For 14 games:
Probability of winning = 60% = 0.6
Amount won = N1,800
Probability of losing = 40% = 0.4
Amount lost = N2,300
Expected return for 14 games = (0.6 * 1,800) - (0.4 * 2,300)
iii. Advising the gambler if he initially staked N1,000:
To calculate the final money the gambler will have after playing the specified number of games, we need to consider the expected returns and the initial stake.
For 12 games:
Expected return for 12 games = (0.6 * 1,800) - (0.4 * 2,300)
Total expected return for 12 games = Expected return for 12 games * 12
Final amount after 12 games = Initial stake + Total expected return for 12 games
For 14 games:
Expected return for 14 games = (0.6 * 1,800) - (0.4 * 2,300)
Total expected return for 14 games = Expected return for 14 games * 14
Final amount after 14 games = Initial stake + Total expected return for 14 games
Based on the calculations, we can advise the gambler on his expected final amount after playing 12 or 14 games.