Probability

evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g. for the total score when two fair dice are thrown), or by calculation using permutations or combinations;

use addition and multiplication of probabilities, as appropriate, in simple cases;

understand the meaning of exclusive and independent events, and calculate and use conditional probabilities in simple cases, e.g. situations that can be represented by means of a tree diagram.

Sample SpaceIt is the list of all possible outcome in an

experiment.

..Choosing a card from a standard pack of

playing cards.The combined experiment of tossing a coin

and a dice.Tossing a drawing pin on to a table to see

whether it lands point down or point up.

A fair no-sided dice has eight faces colored red, ten colored blue and two colored green. Then dice is rolled.

A) Find the probability that the bottom face is red.

B) Let A be the event that the bottom face is not red. Find the probability of A.

Activity 1 Using 2 dice, make a game that has 50 – 50

chances of winning and losing.

The number 1, 2, …, 9 are written on separate cards. The cards are shuffled and the top one is turned over. Calculate the probability that the number on this card is prime.

A circular wheel is divided into three equal sectors, numbered 1, 2 and 3. The wheel is spun twice. Each time, the score is the number to which the black arrow points. Calculate the probabilities of the following

a) Booth score is the same.b) Neither score is a 2.c) At least one of the score is a 2d) Neither score is a 2 and both scores are the same,e) Neither the score is a 2 or both scores are the

same.

Jafar has three playing cards, two queens and a king. Tandi selects one of the cards at random, and returns it to Jafar, who shufffles the cards. Tandi then selects a second card. Tandi wins if both cards selected are kings. Find the probability that Tandi wins?

A dice with six faces has been been made from brass and aluminum and is not fair. The probability of getting 6 is ¼, the probabilities of 2, 3, 4, and 5 are each 1/6, and the probability of 1 is 1/12. The dice is rolled. Find the probability of rolling

a) A 1 or 6.b) An even number.

You draw two cards from an ordinary pack. Find the probability that they are not both kings?

Two pair dice are thrown. A price is won if the total is 10 or if each individual score is over 4. Find the probability that a prize is won?

Weather records indicate that the probability that a particular day is dry is 3/10. Arid F.C. is a football team whose record of success is better on dry days than on wet days. The probability that Arid win on a dry day is 3/8, whereas the probability that they win on a wet day is 3/11. Arid are due to play their next match on Saturday.

a) What is the probability that Arid will win?b) Three Saturdays ago Arid won their match.

What is the probability that it was a dry day?

Conditional ProbabilityA conditional probability refers to the

probability of an event A occurring, given that another event B has occurred.

Notation: P(A B)Read this as the “conditional probability of A

given B” or the “probability of A given B.“These are especially useful in economic

analysis because probabilities of an event differ, depending on other events occurring.

Formulae for conditional probabilitiesThe probability of A given B is

The probability of B given A is

)(

)()(

BP

BAPBAP

)(

)()(

AP

BAPABP

In a carnival game, a contestant has to first spin a fair coin and then roll a fair cubical dice whose faces are numbered 1 to 6. The contestant wins a prize if the coin shows heads and the dice score is below 3. Find the probability that a contestant wins a prize?

A fair of cubical dice with faces numbered 1 t0 6 is thrown four times. Find the probability that three of the four throws result in a 6.

The Special Multiplication Rule (for independent events)

If events A, B, C, . . . are independent, then

P(A & B & C & ¼) = P(A) · P(B) · P(C)¼.

What is the probability of all of these events occurring:

1. Flip a coin and get a head

2. Draw a card and get an ace

3. Throw a die and get a 1

P(A & B & C ) = P(A) · P(B) · P(C) = 1/2 X 1/13 X 1/6

Exercise 4B, numbers 1 -5