Understanding Probability
Every day decisions to some of the most complicated ones are influenced by probability.
In this post, through a game, I discuss probability and associated jargons.
Define the game
Assume a bag consists of 3 RED and 2 BLUE balls. A person is required to pick four balls, one at a time and put the ball back into the bag after noting it’s color. If a person draws four consecutive RED balls, I pay the person $150. If the person does NOT pick four consecutive RED balls, I get $10.Interesting proposition ..eh ! Let’s resist the temptation and not jump right in !
Evaluate the proposition
Find all possible combinations
Find the probability of each combination
Use probability to estimate the chances of winning or losing
List all possible outcomes
We are concerned with the number of RED balls drawn from the bag. This is referred to as a Random Variable which is denoted as ‘X’. Find all possible combinations of the Random variable X
Frequency Distribution Chart
We ask 75 people to carry out the experiment, and record the outcome. Below is a frequency distribution of experiment.
Probability Distribution Table
Given the total number of possible outcomes is 75, the probability distribution is calculated as P(X)/75 .
Probability Distribution Chart
From the probability distribution table, plot the probability distribution chart.
Expected Value
How many red balls does a player pick on an average ? This is answered by finding the Expected value of the experiment.
While the expected value helps me evaluate the average number of red balls picked, it still does not help make a decision i.e what are the chances of a player making or losing money ?
Finally !
To evaluate whether a player is going to lose or make money, I introduce a new random variable, defined as below
X = money won or lost
In this case, X can take 2 probable values i.e +150 or -10. A person wins $150 when 4 consecutive RED balls are drawn and loses ($10) otherwise. We know probability of winning and losing as
Win :: P(4) = 0.1333; Lose :: 1- P(4) = 1- 0.133 = 0.867
Hence the expected value is (0.133* 150) + (0.867 * (-10)) = +11.28
This implies on an average a person playing the game could expect to make $ 11.28.
Conclusion
Looks like the game works in favor of the player ! Given the house always wins, to make the game profitable for the house, the parameters of this game would need to be adjusted.
Decrease the prize money
Increase the penalty
Decrease the chances of winning
