Question

Flip all 3 coins 10 times. Each flip place a tally mark next to the number of heads in the following table

Experiment 1
Number of Heads	TallyMark
0	1
1	4
2	5
3	0
Total Tosses	10

Experiment 2
Number of Heads	TallyMark
0	7
1	17
2	16
3	10
Total Tosses	50

Experiment 3
Number of Heads	TallyMark
0	16
1	41
2	27
3	16
Total Tosses	100

For each of the tables on the previous side, make a frequency distribution and a relative distribution

10 Tosses

50 Tosses

100 Tosses

1. Draw a tree Diagram to represent the event of tossing three coins

2. What is the sample space for this event?                        

3. Using the random variable, X, that measures the number of heads when you toss three coins, write the theoretical probability distribution.

4. Write a few sentences that compare and contrast the relative frequencies between the three experiments. Why are they different or similar?

5. Write a few sentences that compare relative frequencies with the theoretical probabilities. Why would they be similar or different? What happens as the number of trials gets bigger?

Answer 1

Relative frequency:

For first table total frequency = 10, and the frequencies are nothing but the TallyMark

Number of heads	Frequency	Relative frequency
0	1	1/10 = 0.1
1	4	4/10 = 0.4
2	5	5/10 = 0.5
3	0	0/10 = 0

For the second table that is for 50 tosses

Number of heads	Frequency	Relative frequency
0	7	0.14
1	17	0.34
2	16	0.32
3	10	0.2

For the third table that is for 100 tosses

Number of heads	Frequency	Relative frequency
0	16	0.16
1	41	0.41
2	27	0.27
3	16	0.16

1. TREE DIAGRAM :

2. Sample space of three coins:

There are 8 possible outcomes of tossing 3 coins

s = {HHH, HHT, HTT, HTH, THH, THT, TTH, TTT}

3. Theoretical distribution of random variable X that measures the number of heads.

There are 4 outcomes of the number of heads that is 0, 1, 2 and 3

From above sample space, we can write the theoretical probability distribution.

There is 1 outcome of 0 head, 3 outcomes of 1 head, 3 outcomes of 2 head and 1 outcome of 3 heads.

Number of heads (X)	P(X)
0	1/8 = 0.125
1	3/8 = 0.375
2	3/8 = 0.375
3	1/8 = 0.125

This is the theoretical probability distribution.

4. All three experiments of relative frequencies show different results because of the number of trials are different for each experiment.

As the trials increase the relative frequencies increases and then again decreases.

5. As the relative frequencies increase and then decrease at some point, exactly happens about the theoretical probabilities The difference occurs due to the practical and theoretical situation. Because the relative frequencies occur practically so they may change but the theoretical probabilities are same.

If the number of trials gets bigger then the relative frequencies and theoretical probabilities close to each other (that is may same.