Probability

Read these sentences:

• a.The sky is cloudy today; perhaps it will rain.
• b.I am not sure whether I will pass the exam.
• c.The political situation may change suddenly.
• d.The two candidates have an equal chance of winning the election.
• e.Which country is most likely to win the next football World Cup?

Words such as perhaps, not sure, may, chance and likely show uncertainty: the thing may happen, or it may not. We still guess, from what has happened before. Clouds have often brought rain, so we carry an umbrella. People have made such guesses about daily life since ancient times. Probability turns the guess into a number.

The measure of how likely an event is to happen is called its probability. Clouds don’t make rain certain; they make it likely, and probability says how likely, as a number.

Some words of probability

These two examples give us the words we need.

a. Experiment. An action done to see what result it gives: tossing a coin, rolling a die, drawing a card.

b. Random experiment. An experiment whose result can’t be predicted with certainty, although we know every result it could give. Tossing a coin, rolling a die and drawing a lottery are random experiments: nobody can say beforehand who will win the lottery.

c. Outcome. One possible result of a random experiment. When a coin is tossed, H is one outcome and T is the other.

d. Equally likely outcomes. Outcomes that each have the same chance of happening. On the left spinner below, the four colors cover equal areas, so the pointer is as likely to stop on one color as on any other: the four outcomes are equally likely. On the right spinner the colors cover unequal areas, and green, half the board, comes up far more often than red: those outcomes are not equally likely.

e. Sample space. The set of all possible outcomes of a random experiment, written S. For a coin, S = {H, T} and n(S) = 2. For a die, S = {1, 2, 3, 4, 5, 6} and n(S) = 6.

f. Event. A set of some of the outcomes, so a subset of the sample space, usually written E. For a coin the events are {} (neither face, which can’t happen), {H}, {T} and {H, T} (one face or the other, which always happens). An event “happens” when the outcome is one of its members.

g. Mutually exclusive events. Toss a coin and suppose it shows T. Was H showing at the same time? No. Toss it again; it shows H. Could T show too? No. Events that can’t happen at the same time are mutually exclusive: if one happens, the other can’t. Getting H and getting T are mutually exclusive. So are the six faces of a die: one roll shows only one face.

h. Number of favorable outcomes. The number of outcomes that make an event happen, n(E). For “an even number” on a die, the favorable outcomes are 2, 4 and 6, so n(E) = 3.

i. Elementary event. An event with just one outcome. Rolling a die, the elementary events are {1}, {2}, {3}, {4}, {5} and {6}.

The probability of an event

The possible outcomes are S = {H, T}, so n(S) = 2. Getting a head is the event E = {H}, with one favorable outcome, n(E) = 1. One outcome out of two equally likely ones: the probability of a head is $\frac{1}{2}$, and so is the probability of a tail.

When all the outcomes of an experiment are equally likely, the probability of an event E is

$$P(E) = \frac{n(E)}{n(S)}$$

that is, the number of favorable outcomes divided by the number of all possible outcomes. For a head, $P(H) = \frac{1}{2}$.

The rule needs the outcomes to be equally likely. On the unequal spinner above there are 4 colors, but the probability of green is not $\frac{1}{4}$: it is $\frac{1}{2}$, its share of the board. A fair coin, a well-made die, a well-shuffled pack of cards and balls of the same size in a bag drawn without looking all give equally likely outcomes.

The event “not E”, written E and read “E bar”, is every outcome that is not in E: its complement. The outcomes in E and in E together make up all of S, so n(E) + n(E) = n(S). Dividing by n(S):

$$P(E) + P(\overline{E}) = 1, \qquad P(\overline{E}) = 1 - P(E)$$

A pack of playing cards (तास) has 52 cards in four suits of 13: spades (हुकुम) and clubs (चिडी) are black; hearts (पान) and diamonds (इँटा) are red. Each suit has an ace, the numbers 2 to 10, and three face cards, the jack (गुलाम), queen (रानी) and king (बादशाह); the ace is the एक्का. So a pack has 26 red and 26 black cards, 4 aces, 4 kings and 12 face cards.

Suit	Color	Cards in the suit
Spades ♠	black	A, 2–10, J, Q, K: 13
Clubs ♣	black	13
Hearts ♥	red	13
Diamonds ♦	red	13

Question 1

Write the sample space of each experiment.

Question 2

Write the event (the favorable outcomes) in each case.

Question 3

Question 4

Question 5

A family is picked at random from these.

Children in the family	0	1	2	3
Families	9	15	36	30

There are 9 + 15 + 36 + 30 = 90 families, so n(S) = 90.

Question 6

No event can have fewer favorable outcomes than none, or more than all of S: 0 ≤ n(E) ≤ n(S). Dividing by n(S),

$$0 \le P(E) \le 1$$

A probability of 0 means the event is impossible; a probability of 1 means it is certain, as P(S) = 1 always is. Every other event lies in between. The line from 0 to 1 on which probabilities are marked is the probability scale.

Question 1

Question 2

A bag has 2 blue, 3 black and 5 red marbles of the same size. One marble is drawn at random. Here n(S) = 2 + 3 + 5 = 10.

Question 3

Question 4

A card is drawn from a well-shuffled pack of 52 cards.

Question 5

The spinner is divided into four equal parts, yellow, red, blue and green. Find the probability that the pointer stops on each of the following, and show the results on a probability scale.

Four equal parts: n(S) = 4.

Question 6

Question 7

Cards numbered 2 to 25 are shuffled and one is drawn at random. There are 25 − 2 + 1 = 24 cards, so n(S) = 24.

Counting says a coin gives a head with probability $\frac{1}{2}$, so in 10 tosses we expect about $10 \times \frac{1}{2} = 5$ heads, and in 20 tosses about $20 \times \frac{1}{2} = 10$. Try it, and you will often get 4 or 7 heads in 10. A real experiment rarely matches the count exactly.

A probability found by doing an experiment many times is an empirical probability (also called experimental probability):

$$\text{Empirical probability} = \frac{\text{number of times the event happened}}{\text{total number of trials}}$$

This is the event’s relative frequency. The probability from counting, $\frac{n(E)}{n(S)}$, is sometimes called the theoretical probability, to tell the two apart.

Over a few rolls the shares jump about. Over hundreds and thousands they settle near $\frac{1}{6}$: the more trials, the closer the empirical probability usually comes to the theoretical one. That is why the empirical probability is the one we use where counting is impossible: no one can count the “equally likely outcomes” of a brick breaking, a seed sprouting or a baby being a boy, but records of many cases give the probability.

An empirical probability also predicts: in N more trials, the event is expected about N × P(E) times.

Ishwori asked whether a six is rarer than other faces.

No: on a fair die a six has probability $\frac{1}{6}$, exactly the same as a three or any other face.

The widget shows it: after a few thousand rolls the six’s bar sits on the $\frac{1}{6}$ line with the others. Over many rolls a six comes about once in every six rolls. But that is an average. The same widget also keeps the longest wait for a six, and in a few hundred rolls it is nearly always longer than 8 rolls. Eight rolls without a six is unlucky, not strange: in Grade 10 you will work out that it happens nearly one time in four, $\left(\frac{5}{6}\right)^8 \approx 0.23$. And the die is not keeping score: her next roll is a six with probability $\frac{1}{6}$, however long she has waited.

Grade 9’s final exam is set to CDC’s specification grid for Compulsory Mathematics, Grades 9–10. There, Statistics and Probability is 11 marks in two questions. In the grid’s model paper, question 15 is on number cards:

Part	Level	Marks	What it asks
a.	knowledge	1	the formula for P(E)
b.	understanding	2	the probability of an event
c.	application	2	the probability of an event not happening
d.	higher ability	1	when a probability is greatest, with a reason

Here is one of our own in the same shape.

Question 1

Question 2

Question 3

Three coins are tossed together. Write the sample space, then answer.

Question 4

A die is rolled. Write its sample space, S = {1, 2, 3, 4, 5, 6}, and find each probability.

Question 5

A die was rolled 60 times, with these results:

Face	1	2	3	4	5	6
Frequency	8	9	13	15	11	4

Find the empirical probability of each.

Question 6

These questions cover Units 17, 18 and 19.

Question 1

Question 2

Question 3

Question 4

Find the arithmetic mean, median, mode, first quartile Q₁ and third quartile Q₃ of each.

Question 5

A fair six-faced die was rolled 56 times, with these results:

1, 6, 5, 6, 4, 3, 4, 4, 3, 5, 1, 1, 6, 6, 5, 4, 3, 2, 2, 2, 6, 5, 1, 4, 3, 1, 5, 4, 6, 4, 3, 2, 2, 3, 1, 1, 1, 1, 6, 6, 2, 1, 3, 6, 4, 1, 5, 6, 6, 1, 3, 5, 5, 4, 6, 3

Write the data as a discrete series (it is worked once here, in the answer to a), then: