Probability - discrete concepts TYS 1

Random variables.

1. A variable is defined as the number of kilometres a car travels on one tank of petrol.

(a) What are possible values for this variable?

(b) How many numbers could be recorded for the possible values?

(c) Is the variable discrete or continuous?

2. A random variable is defined as the number of people using a pedestrian crossing at a local shopping centre per hour.

(a) What are possible values for this random variable?

(b) Is there a finite number of values for the variable?

(c) Is the variable discrete or continuous? Explain your answer.

3. A teacher gives the same statistics test to several classes of students across different schools. The maximum mark for this test is 42.

(a) What are possible values for the random variable of the mark obtained?

(b) Is there a finite number of values for the random variable?

(c) Is the variable discrete or continuous? Explain your answer.

4. Can each of the following characteristics be classified as being random variables?

(a) car colours.

(b) energy providers.

(c) energy produced by solar panels manufactured by a variety of companies.

(d) the numbers formed by throwing two dice and writing down the upmost values in order.

6. Which of the following is NOT an example of a discrete random variable?

(a) The number of students sitting for Mathematics over a number of years.

(b) The number of pens found in your bag before an exam.

(c) The number of teachers in the Maths staffroom at lunch time.

(d) The lengths of the songs on your play-list.

7. Indicate for each of the following if the distribution described is a discrete probability distribution? Explain each of your answers?

(a) the weight of each member of a class.

(b) the results of throwing a dice.

(c) the number of glasses of water a person drinks in a day.

(d) the age of each passenger in a bus.

Estimate probabilities.

8. What is the value of a in the following table if the Pr (X = x) is a discrete probability distribution?

x	-2	-1	0	1	2
Pr (X=x)	0.0	0.4	a	0.0	0.5

Answer.As the probabilities must add to 1.0, a = 0.1.

9. Explain why the following table is not a discrete probability distribution for Pr (X = x).

x	10	11	12	13	14
Pr (X=x)	0.1	0.2	0.3	0.0	0.5

Answer.The probabilities do not add to 1.0.

10. A probability distribution is summarised as:

x	1	2	3	4
Pr (X=x)	k	2k	3k	k

What is the value of k?

Answer.k = 1/7.

11. The distribution of a discrete random variable X with a mean of 2.1 is shown on the table:

x	1	2	3
Pr (X=x)	0.2	a	b

Find the values of a and b.

Answer.a = 0.5, b = 0.3.

12. The School Library keeps a record for each term of the number of weeks books are overdue.

The situation for Term 3 is summarised in the following table:

No. of weeks overdue.	No of students with overdues.
1	100
2	40
3	10
Total	150

Develop the probability distribution of the random variable defined as the number of weeks library books are overdue.

13. The local branch of a Bank monitors the number of people on their books are have not been able to meet payments on their credit card. After 5 months, the records go to Head Office.

The situation at the branch in May is summarised in the following table:

No. of months in arrears.	No of customers not meeting payments.
1	150
2	80
3	50
4	30
5	10
Total	320

Develop the probability distribution of the random variable defined as the number of months customers are not meeting their payments.

14. Let the discrete random variable X be the number showing at the top of a six sided die when it is tossed.

(a) Express the probability distribution of X in tabular form.

(b) Present the probability distribution in graphical form.

15. Let the discrete random variable X be the number of heads when a fair coin is tossed three times.

(a) Express the probability distribution of X in tabular form.

(b) Present the probability distribution in graphical form.

16. The probability distribution of a random variable X is summarised in the following table.

X	-4	0	1	2
p(x)	0.2	0.3	0.4	0.1

Find the following probabilities:

(a) Pr (X > 0).

(b) Pr (X < 0).

(c) Pr (0 ≤ X ≤ 1).

(d) Pr (X = -2).

(e) Pr (X < 2).

17. A probability distribution is defined for the random variable X as being p(X) = kx with the values for x being 1, 2, 3, 4, ... 2n.

(i) Find the probability of selecting an odd number.

(ii) find Pr (X ≤ n).

(ii) Find the value of k.