Question

Answer the following questions thoroughly. Use correct grammar and punctuation.

1) A student observes the following spinner and claims that the color red has the highest probability of appearing, since there are two red areas on the spinner. What is your reply? Your answers should include facts, rules, any definitions necessary to explain why the student is correct or incorrect.

2) In class, an experiment of flipping a coin is performed.

Pretend that you are introducing the concept of probability and introducing the terms 1) experiment, 2) outcomes, 3)sample space, 4) theoretical probability, and 5) empirical probability. Explain these terms, as if you were speaking to a 5th grade class, as if they knew nothing about probability. You will need to tell the class what each term , 1-5, is represented by in this particular example. Use correct grammar and punctuation.

Answer 1

1) Since no information has been provided about the spinner, hence we would assume the spinner to have a total of 5 areas.

Out of those 5 areas, 1 area is colored green, 1 area is colored yellow, 1 area is colored blue and 2 areas are colored red.

Now, if we rotate the spinner any of the 5 areas can appear.

Now, the definition of probability says that it is the measure of the likelihood of the happening of any event and it's value is given by the definition

P = Favorable space / sample space

Sample Space is defined as the number of all possible outcomes and Favorable space is the number of outcomes that we want.

Now, in the above situation, we have 5 areas which can appear. Hence, our size of the sample space is 5.

Hence, now for red color, we have 2 areas which are red. Hence, for red color, our favorable space is of size 2.

For rest of the colors, since they have only 1 area to themselves hence, their favorable space is of size 1.

So, the probability that red color will appear will be equal to = 2/5 = 0.4

The probability that blue will appear = 1/5 = 0.2

The probability that yellow will appear = 1/5 = 0.2 and

The probability that green will appear = 1/5 = 0.2

As we can see, the probability of the appearance of Red is = 0.4 is the highest among all other probabilities. Hence, the student is correct.

2) 1) Experiment - It is the activity that can be performed an infinite number of times and has a fixed set of outcomes. Here, we are flipping a coin. We can flip it an infinite number of times and it has only two outcomes, either heads or tails.

2) Outcomes - These are the possible results of an experiment. In flipping a coin, the outcome can be either a head or a tail. Hence, there are 2 outcomes.

3) Sample space - It is the set of all possible outcomes. In flipping a coin we have two outcomes, head or tail.

Hence the sample space is {Heads, Tails}.

4) Theoretical Probability - It is the likelihood of something to occur. For example, in a coin toss, the likelihood of the result of the toss to be heads will be the theoretical probability of the outcome to be heads.

Theoretical probability is given by

= Favorable space / sample space

In a coin toss, the probability of heads will be = 1/2 = 0.5

and the probability of tails will be = 1/2 = 0.5

5) Empirical probability - This is an experimental probability. We conduct an experiment some number of times and note the outcome. Then we count the number of some particular outcome and divide it by the number of experiments to get the empirical probability of that outcome.

For example

Lets say we toss a coin 3 times. Then we are conducting the experiment 3 times.

Let the outcomes be

H , H and T.

Count the number of heads = 2

Count the number of tails = 1

The empirical probability of getting heads = Number of heads/ Number of trials = 2/3

The empirical probability of getting tails = Number of tails/ Number of trials = 1/3

As we can see, the empirical and theoretical probability is different. The theoretical probability won't change ever but the empirical probability can change since it depends on the outcome of the experiment.

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