Probability | Mathematics homework help

Digging in to Probability

Bernoulli’s Theorem: If an experiment is repeated a large number of times, the experimental probability of a particular outcome approaches a fixed number as the number of repetitions increases. (p.519)

1. Restate Bernoulli’s Theorem in your own words (aimed at the level of an average nine year old).

2. If you have flipped a fair coin 99 times and gotten tails 79 of those 99 times, what is the chance that the next flip of the coin will come up tails? Why?

Experiment: Roll two distinguishable regular four-sided dice. The sides of each die are labeled 1,2,3, and 4. (You do not have to physically roll the dice. You will use this experiment for the rest of the worksheet.)

3. List all elements of the sample space. (Don’t use a tree diagram unless you have to.)

4. Let G be the event that at least one die has a 3. List out the elements of event G. What do you think the probability of event G (written P(G)) is?

5. Let H be the event that both dice have even numbers. List out the elements of event H. What do you think P(H) is?

6. Let I be the event that the sum of the dice is 5. List out the elements of event I. What do you think P(I) is?

7. What is P(G U H)? What is P(G U I)? What is P(H U I)?

Two events are mutually exclusive if they have no elements in common. In other words, their intersection is empty.

8. Look at G;H; and I.

(a) Are G and H mutually exclusive?

(b) Are G and I mutually exclusive?

(c) Are H and I mutually exclusive?

9. Make a conjecture about P(A U B) for any generic events A and B that are mutually exclusive.

10. What is P(A U B) if A and B are NOT mutually exclusive?

11. Let J be the event that at least one die has an odd number. List out the elements of J. Which of G, H, and I is mutually exclusive to J? What is the probability of the union of that event with J?

Two events are complementary if they are mutually exclusive and P(A U B) = 1.

12. Without the concept of probability describe in your own words what it means for two events to be complementary?

Introduction to Probability

Probabilities are ratios. These ratios can be expressed as fractions, decimals, or percents. As the book says, these ratios are based on “outcomes of experiments”. But what does this mean? We will explore the idea with an example.

Example: You have 4 colors of socks in your drawer: white, black, red, and gray. You reach in and pull out a sock.

1. What are the possible outcomes of your experiment?

The set of all possible outcomes is called the Sample Space.

Often times, a sample space can be represented with a tree diagram.

2. What is the sample space if you pull out one sock, record its color, put it back, and then pull out another sock and record its color?

3. Choose a few outcomes from your sample space in question (2).

Any subset of a sample space is called an Event.

4. If there are exactly two of each color of sock, what are the chances of your chosen event from (3) happening?

That is the idea of probability.

Your Turn!

Take a 6-sided dice.

5. You are going to roll the die 12 times, but before you do answer the following question. How many times would you expect it to land with the 3 up and how many times would you expect it to land with the 5 up?

6. Roll the die 12 times. Record your results for all 12 rolls.

7. Did this match your prediction from question (5)? If not, why not?

Empirical/Experimental versus Theoretical Probability

When a probability is determined by observing outcomes of experiments it is Experimental (empirical) probability.

When a probability is determined by the limit of outcomes of more and more and more experiments it is _________________________ probability.

8. What was the empirical probability of your die experiment? What was the theoretical probability?

9. In general, what numbers represent the lowest and highest possible probability? Why?

For our purposes “probability” with no adjective before it means theoretical probability.

Conditional Probability is:

Example #1: You pull one card out of a standard deck of cards.

1.      What is the chance that the card is a king?

1/13

2.      If you know that your card must be a face card, what is the chance that it is a king?

12/52 which is 3/13

3.       If you know that your card must be a king, what is the chance that it is a face card?

12/52 which is 3/13

Let F be the event that you pull a face card and K be the event that you pull a king. For the statement in question 2 above, we say what is the chance of  “K given F.”

The notation for this is:

What would the notation be for question 3 above? What would the English translation be?

Example #2: You roll a fair six sided die. Let Q be the event that you rolled a prime number. Let D be the event that you rolled a number less than 5. Find the following:

P(Q)

P(D)

P(Q|D)

P(D|Q)

Example #3: You roll a fair four sided die twice. Let A be the event that you roll a 3 on the first roll and B be the event that the sum of the two rolls is 4. Find the following:

P(A)

P(B)

P(B|A)

P(A|B)