Probability Examples
1. You apply to two graduate schools: Harvard and Pitt. The probability that you get accepted into Harvard is 0.3, the probability you get into Pitt is 0.7, and the probability you get accepted into both Harvard and Pitt is 0.25 Are the two individual events – getting accepted into Harvard and getting accepted into Pitt – considered independent events? Why or why not?
Dependent because P(H)*P(P) = P(H and P) that is: 0.3*0.7 = 0.21 which does not equal 0.25
2. You have a six-sided die consisting of green on four sides and red on the other two. You roll the die five times. Which of the following sequences is MOST likely to happen?
Green, Green, Green, Green, Green
Green, Red, Green, Red, Green
Green, Green, Green, Red, Green
The sequence with the most green since green is most likely to appear after each toss.
3. The football game is almost over. Your team has the ball and is winning by four points meaning the other team needs a touchdown to beat you. It is fourth down and you need two yards for a first down. You have two choices:
1. Go for the first down. If you are successful you win. The chances of getting the first down are 60%. If the play fails, then the other team gets the ball with a 53% chance of scoring the winning touchdown.
2. Or you can kick the ball to the other team. By choosing this option your team has a 70% chance of winning.
Which choice should you make? That is, which of the two options results in the greater probability that your team wins?
Probability win by going for if kick is 0.70 so you would not kick if probability of winning for this choice exceeds 0.70
Probability of win if no kick is found by:
(0.6)*(1) + (0.4)(1-0.53) = 0.6 + 0.188 = 0.788
Since probability of winning for not kicking is greater than probability of winning if kicked, the decision would be to not kick.
4. Back in the 1970s there was a game show called "Let's Make a Deal". The premise of the game was this: A contestant was shown three curtains behind one of which was a real prize (e.g. money, new car) while the other two hid gag prizes (e.g.

donkey eating hay). The rules were quite simple. The contestant picked a door (e.g. curtain #1) and the host then opened one of the other curtains behind which was a gag gift (e.g. curtain 3). This left the contestant with two curtains still remaining; the one chosen and the other. The host then offered the contestant the option to either stick with their original curtain or switch to the other. Question: If you were the contestant, would you switch at this stage of the game or stay with your first choice? That is, which decision - staying with first choice or switching to other curtain – gives you a better chance of winning?
You Pick Prize Door Monty Opens
Don’t Switch
Switch
1 1 2 or 3 Win Lose
1
1
2
3
3
2
Lose
Lose
Win
Win
2
2
2
3
1
2
3
1
3
1 or 3
1
2
Lose
Win
Lose
Lose
Win
Lose
Win
Win
3
3
2
3
1
1 or 2
Lose
Win
Win
Lose
3 Wins (33%) 6 Wins (66%)