Download Form groups of two or three and discuss the following questions

Module H1 Practical 1
Basic Probability Ideas
Getting together in pairs, discuss the following questions. After discussion some groups
will be randomly chosen to present the answer to one question to the rest of the class.
1(a) Suppose an outcome from a random event has a probability 0.02 of occurring. Which
of the following statements represents the correct interpretation of this probability?
(i)
The outcome will never happen.
(ii) The outcome will certainly happen two times out of every 100 trials.
(iii) The outcome is expected to happen about two times out of every 100 trials.
(iv) The outcome could happen, or it couldn't, the chances of either result are the same.
(b) The Hai Meteorological Center wanted to determine the accuracy of their weather
forecasts. They searched their records for those days when the forecaster had reported a
70% chance of rain for the following day. They compared these forecasts to records of
whether or not it actually rained on the day for which the forecast was intended. Indicate
which of the following statements is correct.
The forecast of 70% chance of rain can be considered very accurate if it rained on:
(i)
95% to 100% of those days.
(iv)
65% to 74% of those days.
(ii)
85% to 94% of those days.
(v)
55% to 64% of those days.
(iii)
75% to 84% of those days.
(c) Tendai, a new statistics graduate was asked to estimate the probability of rain on June
16, a day on which an important national celebration was to take place. He used
meteorological data of the last 70 years to obtain the number k of June 16 days in which it
rained. He presented his estimate of probability of rain as k/70. Which of the three
methods did Tendai use? Explain.
(i)
The subjective (Bayesian) approach;
(ii)
The classical approach;
(iii) The frequentist approach.
SADC Course in Statistics
Module H1 Practical 1 – Page 1
Module H1 Practical 1
2. Suppose two six-sided dice are thrown simultaneously.
(a) Complete the following table showing all possible events that can occur. Here (i, j) in
the table cells represents the number i appearing uppermost on the first die and the
number j appearing uppermost on the second die.
Values j on second die
Values i
on first
die
1
1
2
3
(1,1)
(1,2)
(1,3)
2
(2,1)
(2,2)
(2,3)
4
5
6
3
4
5
6
(b) What is the probability that
(i)
the sum of the numbers on the two dice will be 5 or less;
(ii)
the number j on the second die will be 5 or more;
(iii) both dice will show the same number.
(c) In computing the above probabilities, which approach to probability are you using? Is
it the subjective, classical or frequentist approach? Justify your answer.
SADC Course in Statistics
Module H1 Practical 1 – Page 2
Module H1 Practical 1
3. Some components of a simple life table are shown below. The final column of this
table gives the probability of a person dying in the given age group. Remember from
intermediate module I3, session 14, that:- “The number who die between exact ages x and
(x+1) divided by the total number who lived to age x, is denoted qx. It is the estimated
probability of dying aged x”. In demographic jargon this is called the age-specific mortality rate.
Note that in this example we have mortality rates for some wider age-ranges.
Age range
Survivorship at start
of age group
Probability of
dying
<1
100000
0.05465
1-4
94535
0.01906
5-9
0.00877
10-14
0.00604
15-19
0.01306
20-24
0.03161
25-29
0.04639
30-34
0.08196
35-39
0.13478
40-44
0.13074
45-49
0.12092
50-54
0.14089
55-59
0.15467
60-64
0.18425
65-69
0.23618
70-74
0.31338
75-79
0.42484
80-84
0.56888
85-89
0.72866
90-94
0.81936
95-99
0.86743
100+
1.00000
SADC Course in Statistics
Module H1 Practical 1 – Page 3
Module H1 Practical 1
If there are 100000 live births, in the population to which the above results apply, discuss
the following in small groups of 3-4 persons. Write down your reasoning in each case:(i) how the second number in the “Survivorship” column is derived from those in the
“Age < 1” row;
(ii) how to calculate the third and subsequent numbers in the “Survivorship” column.
Note that the solution will be discussed in full during Session 11.
(iii) After being born alive into this population, can you work out what the probability is of
surviving from birth to exact age 5?
Note that the solution should be clearer after Session 3.
4. You will meet mortality rate schedules like the above on several future occasions in this
module. By hand or using the computer, produce a graph of “Probability of Dying” vs.
“Age” ensuring this is accurately labelled and takes account of the differing widths of the
age groups shown.
From the graph, try estimate the later year of age where the one-year probability of dying
first exceeds that for the year from birth to age 1.
SADC Course in Statistics
Module H1 Practical 1 – Page 4