Random Experiments & Outcomes

Introduction:

Statistics and Probability concern themselves with measuring things then predicting results about the measured things. For example, politicians take polls to predict the outcome of an election, traffic directors count vehicles at busy intersections to set traffic light sequences and decide where to build new roads. Gamblers count cards or figure out the number of ways to make 7 or 11 using a pair of balanced dice. As a matter of fact, it was gamblers who began developing this branch of mathematics in order to predict the outcomes of the games they played such as roulette and blackjack.

As in all branches of mathematics, statistics uses a precise vocabulary to define and discuss the measuring and predicting processes as well as the thingies we measure and predict about.

Terms and Definitions

In statistics, the terms: experiment, outcome, event and probability describe what we do, what we get and the likelihood of getting it.

experiment: an activity or act of measurement (tossing a coin), where there are 2 or more possible and uncertain results (Heads or Tails).

outcome: a particular result of an experiment (getting Heads)

event: a collection of one or more outcomes of an experiment.

probability: a number greater than or equal to 0, less than or equal to 1 that measures the likelihood a specific event will occur.

So, if we toss a coin there are 2 possible outcomes: Heads or Tails. And if we call Heads a successful result, the experiment is tossing the coin, the outcome we want is Heads, so the probability of a successful outcome is ½ or 50%, since there are only 2 possible outcomes.

The probability of a given outcome is the number of successes (Heads)
divided by the number of possible outcomes (Heads or Tails).

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Random Experiments

A random experiment is a simple act of measurement such as tossing a coin, rolling a die, spinning a roulette wheel, guessing a number between 1 and 10 etc. Experiments can be simple or compound.

A Simple Experiment involves one activity such as tossing a coin OR spinning a wheel.

A Compound Experiment involves 2 or more acts like tossing a coin AND spinning a wheel.

Counting Possible Outcomes in a Random Experiment

The set of outcomes in a random experiment includes all the possible results of the experiment.

If we roll a balanced die, there are 6 possible outcomes: 1, 2, 3, 4, 5 or 6.

If we pick a winner from Bob, Joe, and Pam, there are 3 possible outcomes:
Bob wins, or Joe wins or Pam wins.

Now say we first flip a coin and then spin the pointer on the four color wheel. Since we can get either Heads (H) or Tails (T) from the coin toss and Red, Blue, Green or Yellow from the wheel, there are 8 possible outcomes from this compound experiment. They are:

(H, Blue), (H, Red), (H, Green) and (H, Yellow)
(T, Blue), (T, Red), (T, Green) and (T, Yellow)

The total number of outcomes from a compound experiment in which
the first activity has n possible outcomes and
the 2nd activity has m possible outcomes is mn.

So, our coin flip and wheel spin has 2 % 4, or 8 possible outcomes.

Say you have 4 pairs of slacks and 3 warm shirts. You have a total of 12 different shirt and slacks combinations to choose from when you get dressed.

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Tree Diagrams

Sometimes it is best to illustrate the set of possible outcomes from a compound experiment with a tree diagram to display all the possible results of the situation in question.

example: say we're judging an essay writing contest and we've narrowed the field down to 4 finalists: Pam, Sam, Harry and Barry, from which we must pick the first and 2nd prize winners. Here is the tree diagram showing all the possible ways we can pick the 2 winners from the 4 finalists.

We see there are 12 possibilities. Once we pick a 1st prize winner, we have 3 people left from which we choose the 2nd prize winner. So, there are 4 choices for 1st prize and three 2nd prize candidates for each of those 4 for a total of 4 % 3 or 12 possible choices for these 2 prizes.

In this experiment the probability that any 2 people get the 1st and 2nd prize is 1/12.

Another way to do it is to imagine we have the four finalists and 2 chairs, one marked 1st prize and the other marked 2nd prize. We have 4 ways to fill the 1st prize chair, then we have 3 ways to fill the 2nd prize chair, so we have 12 ways to do it in all.

example: A student can play 0, 1, or 2 hours of raquetball on any given night.
Construct a tree diagram to determine the number of ways that in 3 nights
s/he can play for a total of 4 hours (total hours over the 3 nights = 4).

There are 6 ways to play for a total of 4 hours over 3 nights.

example: We have a red die and a blue die. We want to list all possible outcomes of rolling these dice with the red one first and the blue one second.

The set of all outcomes includes getting 1, 2, 3, 4, 5 or 6 on each die so it is:

(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

There are 6 % 6 = 36 possible outcomes (ordered pairs) of this experiment.

Now say we define event A as the set of all outcomes with equal values on both dice.
This set includes (1, 1) (2, 2) (3, 3) (4, 4) (5, 5) and (6, 6).
Therefore, the probability of rolling doubles is 6/36 or 1/6.

Now let's list the outcomes in event B in which the sum of the two values = 5
This set includes (1, 4) (2, 3) (3, 2) (4, 1).
Therefore, the probability of rolling a five is 4/36 or 1/9.

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Practice

1) A café offers 5 types of sandwich, 4 kinds of juice, and 6 side dishes as it's noon special. How many different meals can be made from the items on special?

2) Kristen can travel from Montreal to Toronto by bus, train or plane and she can get to Winnipeg from Toronto by bus or by train. In how many different ways can Kristen get from Montreal to Winnipeg?

3)
a) Using H for Heads and T for Tails, list all the possible outcomes of tossing 3 coins.
b) How many outcomes are there?
c) What is the probability of rolling 3 Heads?
d) What is the probability of rolling 2 Heads and 1 Tail?

4) The experiment is tossing a coin (H or T) and rolling a die.
a) How many possible outcomes are there for this experiment?
b) List all the possible outcomes for this experiment.
c) List the outcomes in the event that the coin comes up Tails.
d) List the outcomes in the event that the die shows an even number.
e) What is the probability of getting Heads and an odd number?

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Solutions

1) There are 5 % 4 % 6 or 120 different meals that can be created from the specials.

2) Kristen can travel in 3 % 2 or 6 different ways from Montreal to Winnipeg.

3) a)

 H H H H H T H T H T H H H T T T H T T T T T T H

b) There are 2³ or 8 possible outcomes.
c) The probability of rolling 3 Heads is 1/8 since it happens only once in 8 outcomes.
d) The probability of rolling 2 Heads and 1 Tail is 3/8 since we see H H three times.

4 a) Since there are 2 choices for the coin and 6 for the die,
there are 12 equally possible outcomes.
b) The 12 equally possible and probable outcomes are:

 H 1 H 2 H 3 H 4 H 5 H 6 T 1 T 2 T 3 T 4 T 5 T 6

c) the answer is the bottom row of the table above.
d)
 H 2 H 4 H 6 T 2 T 4 T 6

e) The probability of getting Heads and an odd number is 3/12 or ¼.

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