Comparing Probabilities

How can you tell which is experimental and which is theoretical probability?

You tossed a coin 10 times and recorded a head 3 times, a tail 7 times

P(head)= 3/10P(tail) = 7/10

Toss a coin and getting a head or a tail is 1/2.

P(head) = 1/2P(tail) = 1/2

Remember: Theoretical Probability is what “should” happen and Experimental Probability is what actually happened during an experiment.

Experimental: Theoretical:

More on Theoretical Probability

How do you figure out theoretical probability?Use the sample space (all possible outcomes) and what we

know about the event. How many possibilities are there total?Is it equally likely that all outcomes can occur?

More on Experimental ProbabilityUsing results from an experiment (observed

relative frequencies) is another method to estimate the probabilities of a given event

Relative frequency - the proportion you get from the number of the observed outcomes of an event divided by the total number of outcomes. Relative frequency can be expressed as a fraction, a

decimal, or a percent. Think back to yesterday’s activity, what was the

frequency of the coin landing on heads?

Is the estimate of a probability found through an experiment close to the theoretical probability?

Beth tosses a coin 10 times and records her results. Here are the results from the 10 tosses:

What is the relative frequency of heads?Hint: It is the proportion of the time that heads occurred on

these tosses.

What is the relative frequency of tails?

What is the sum of relative frequencies of heads and tails?

Toss

Result H H T H H H T T T H

total number of tails total number of tosses

total number of heads total number of tosses

¿410

=25

35+25=55 All results will add up to 1 every time!

Identifying the Type of Probability

A bag contains three red marbles and three blue marbles.

P(red) = 3/6 =1/2What type of Probability? Theoretical (The result is based on the

possible outcomes)

Identifying the Type of Probability

Trial Red Blue

1 1

2 1

3 1

4 1

5 1

6 1

Total 2 4

Exp. Prob. 1/3 2/3

You draw a marble out of the bag, record the color, and replace the marble. After 6 draws, you record 2 red marbles

P(red)= 2/6 = 1/3Experimental(The result is found by

repeating an experiment.)

What if you don’t know the Theoretical Probability?

Can you use the Experimental Probability to help you instead?If you repeat the experiment many times (long-run

relative frequencies), the results will getting closer to the theoretical value.

Use probabilities to determine whether or not the simulation supported a given theoretical probability. The more trials that are observed from a simulation, the better.

Practice