Probability
Define probability
The extent to which something is probable; the likelihood of something happening or being the case
Define conditional probability
The probability of an event (A), given that another (B) has already occurred
The likely outcome of a scenario
What do simulations model?
Random events
In a normal distribution, data is __1__ distributed with no __2__.
1: symmetrically
2: skew
What is the probability formula?
P(event)= # of favorable outcomes / # of possible outcomes
What is the formula for conditional probability?
P(A|B) = P(A∩B) / P(B)
You take out a fire insurance policy on your home. The annual premium is $300. In case of fire, the insurance company will pay you $200,000. The probability of a house fire in your area is 0.0002.
A) What is the expected value?
B) What is the insurance company’s expected value?
C) Suppose the insurance company sells 100,000 of these policies. What can the company expect to earn?
A) Expected value = (0.0002)(199,700) + (0.9998)(−300) = − $260.00. The expected value over many years is −$260 per year. Of course, your hope is that you will never have to collect on fire insurance for your home.
B) The expected value for the insurance company is the same, except the perspective is switched. Instead of − $260 per year, it is +$260 per year. Of this, the company must pay a large percent for salaries and overhead.
C) The insurance company can expect to gross $30,000,000 in premiums on 100,000 such policies. With a probability of 0.0002 for fire, the company can expect to pay on about 20 fires. This leaves a gross profit of $26,000,000.
True or False:
A simulation is an experiment.
True
The mean of a normally distributed set of data is 56, and the standard deviation is 5. In which interval do approximately 95% of all cases lie?
A) 46-56
B) 51-61
C) 46-66
D) 56-71
C) 46-66
What type of event is the addition rule of probability?
A mutually exclusive event: P(A or B) = P(A) + P(B)
P(A or B) = 0
A card is drawn at random from a standard deck of cards. Recall that there are 13 hearts, 13 diamonds, 13 spades and 13 clubs in a standard deck of cards.
A) If I draw a card at random from the deck of 52, what is P(H)?
B) If I draw a card at random, and without showing you the card, I tell you that the card is red, then what are the chances that it is a heart?
C) If I draw a card at random from the deck of 52, what is P(F)?
A) 13/52 = 25%
B) 13/26 = 50% Because you know that the card drawn is red, the Sample Space of all possible outcomes has size 26 (there are 26 red cards), not 52. Of the 26 red cards, 13 are hearts, so there are 13 successful outcomes.
C) There are 12 face cards so P(F) = 12/52 = 3/13, which is about 0.23
You draw one card from a standard deck of playing cards. If you pick a heart, you will win $10. If you pick a face card, which is not a heart, you win $8. If you pick any other card, you lose $6.
Let X be the random variable that takes on the value s 10, 8 and –6, the values of the winnings. First, we calculate the following probabilities:
P(X=10) = 13/52, P(X=8) = 9/52, and P(X=-6) = 30/52
The expected value is:
E(X) = P(X=10) 10 + P(X=8) 8 - P(X=-6) 6
= 13/52 10 + 9/52 8 - 30/52 6
= 130 + 72 - 180 / 52
= 22/52
Since the expected value of the game is approximately $42, it is to the player's advantage to play the game.
Jessie decided to conduct an experiment with a spinner. The spinner is divided into four colors: red, blue, orange and green. Jessie predicted that out of 30 spins that the spinner would be red 10% of the time.
She conducted the experiment and the spinner was red four times. Is her prediction correct?
First, to figure this out, you must first write a probability and then compare it to the 10% that Jessie predicted. The actual result is that the spinner was red 4 out of 30 times.
P(red)=4/30
Second, convert to a percent.
P(red)=4/30
P(red)=0.133
P(red)=13.3%
The answer is 13.3%.
Jessie’s prediction was too low. The actual result was higher than 10%.
The amount of juice dispensed from a machine is normally distributed with a mean of 10.50 ounces and a standard deviation of 0.75 ounce. Which interval represents the amount of juice dispensed about 68% of the time?
A) 9.00-12.00
B) 9.75-11.25
C) 9.75-10.50
D) 10.50-11.25
B) 9.75-11.25
How do you know if both A & B are independent events?
If the laws match up within P(A and B)
On each day I recorded, whether it was sunny, (S), or not, (NS), and whether my mood was good, G, or not (NG).
S NS
G 9 6
NG 1 14
A) If I pick a day at random from the 30 days on record, what is the probability that I was in a good mood on that day, P(G)?
B) What is the probability that the day chosen was a Sunny day, P(S)?
A) The sample space is the 30 days under discussion. I was in a good mood on 9 + 6 = 15 of them so P(G) = 15/30 = 50%
B) The sample space is still the 30 days under discussion. It was sunny on 9 + 1 = 10 of them so P(S) = 10/30, which is about 33%
You pay $10 to play the following game of chance. There is a bag containing 12 balls, five are red, three are green and the rest are yellow. You are to draw one ball from the bag. You will win $14 if you draw a red ball and you will win $12 is you draw a yellow ball. How much do you expect to win or loss if you play this game 100 times?
Here, the gross winnings are 14, 12, or 0. Since you must pay $10 to play, the net winnings are 4, 2, and –10. Let X be the random variable that takes on the values 4, 2, and –10, the values of the net winnings.
P(X = 4) = 5/12, P(X = 2) = 4/12, and P(X = -10) = 3/12
The expected value of the game is given by:
E(X) = 4 * 5/12 + 2 * 4/12 - 10 * 3/12
= 20 + 8 - 30 / 12
= -2/12, simplified it will be -1/6
True or False:
To use a coin in a simulation, you would need to flip the coin many, many times.
True
In a NYC high school, a survey revealed the mean amount of cola consumed each week was 12 bottles and the standard deviation was 2.8 bottles. Assuming the survey represents a normal distribution, how many bottles of cola per week will approximately 68% of the students drink?
A) 6.4-12
B) 9.2-14.8
C) 6.4-17.6
D) 12-20.4
B) 9.2-14.8 bottles of cola per week
There are 6 blueberries, 4 grapes, and 5 raspberries in a bowl. What is the probability of selecting a blueberry on the first draw?
6/15
A family has two children. Assuming that boys and girls are equally likely, determine the probability that the family has…
A) One boy and one girl GIVEN the first child is a boy.
B) Two girls GIVEN that at least one is a girl
C) Two girls GIVEN that the older one is a girl
A) 1/2
B) 1/3
C) 1/2
A detective figures that he has a one in nine chance of recovering stolen property. His out of-pockets expenses for the investigation are $9,000. If he is paid his fee only if he recovers the stolen property, what should he charge clients in order to break even?
In this problem we want to determine the detective’s fee so that the expected value is zero. Let y be the amount of his fee. Let X be the random variable that takes on the values y or 0, the amount he charges for a job. Then P(X = y) = 1/9 and P(X = 0) = 8/9.
The detective is out $9,000 regardless of whether he recovers the stolen property. So we have
E(X) = (y - 9000) * P(X = y) - 9000 * P(X =0)
= (y - 9000) * 1/9 - 9000 * 8/9
Solving for y we see that the detective must charge $81,000.
In order to obtain the probability of various events, a simulation is often conducted. It allows us to represent the __1__ of a real-life event occurring by using an experiment with __2__.
1. likelihood
2. similar probabilities.
On a standardized test, the mean is 76 and the standard deviation is 4. Between which two scores will approximately 68% of the scores fall?
A) 68 and 84
B) 72 and 80
C) 74 and 78
D) 76 and 80
B) 72 and 80