Class 10 Mathematics · Chapter 14 NotesProbability

In Class IX you met experimental or empirical probability, which is based on what actually happens when an experiment is repeated many times. It is calculated as the number of trials in which the event happened divided by the total number of trials. This works well for tossing coins or throwing dice, but repeating an experiment can be expensive or impossible, such as launching a satellite again and again. When we are willing to make assumptions, we can avoid repetition and calculate the exact theoretical probability. The key assumption is that all outcomes of the experiment are equally likely, meaning no outcome has a better chance than another. The theoretical probability of an event E, written P(E), is the number of outcomes favourable to E divided by the number of all possible outcomes of the experiment. This definition was given by Pierre Simon Laplace in 1795.

• Experimental probability P(E) = number of trials in which the event happened / total number of trials
• Theoretical probability P(E) = number of outcomes favourable to E / number of all possible outcomes
• Theoretical probability assumes equally likely outcomes

A fair coin is symmetrical, so there is no reason for it to land more often on one side than the other. We call such a coin unbiased, and we say that head and tail are equally likely outcomes. Similarly, a fair die has six faces, and the outcomes 1, 2, 3, 4, 5 and 6 are equally likely. However, not every experiment has equally likely outcomes. If a bag contains 4 red balls and 1 blue ball, drawing a red ball is more likely than drawing a blue ball, so the outcomes 'red ball' and 'blue ball' are not equally likely. But drawing any one particular ball from the bag is equally likely. In this chapter, unless stated otherwise, we assume that the outcomes of the experiments we study are equally likely.

• Fair coinhead and tail are equally likely
• Fair die1, 2, 3, 4, 5, 6 are equally likely
• Outcomes of an experiment need not always be equally likely

An event having only one outcome of the experiment is called an elementary event. For example, when a coin is tossed once, getting a head is an elementary event, and getting a tail is another elementary event. When a die is thrown once, getting 1, getting 2, and so on up to getting 6 are all elementary events. The complete list of all possible outcomes of an experiment is called the sample space. For tossing one coin it is {H, T}; for throwing one die it is {1, 2, 3, 4, 5, 6}. When two coins are tossed simultaneously, the possible outcomes are (H, H), (H, T), (T, H) and (T, T). When two dice are thrown, the outcomes are ordered pairs such as (1, 1), (1, 2), and so on, giving 6 × 6 = 36 outcomes. The pair (1, 4) is different from (4, 1) because the two dice are different.

• Elementary eventan event with only one outcome
• Sample spacethe set of all possible outcomes
• Two coins4 outcomes; two dice: 36 outcomes

To find the probability of an event, first count the total number of equally likely outcomes, then count how many of them are favourable to the event, and divide. For example, when a die is thrown once, the probability of getting a number greater than 4 is 2/6, because only 5 and 6 are favourable out of six outcomes. The probability of getting a number less than or equal to 4 is 4/6. When a card is drawn from a well-shuffled deck of 52 cards, the probability that it is an ace is 4/52, since there are 4 aces. The sum of the probabilities of all the elementary events of an experiment is always 1. For instance, in a bag with 3 blue, 2 white and 4 red marbles, P(white) = 2/9, P(blue) = 3/9 and P(red) = 4/9, and these add up to 1.

• P(E) = number of outcomes favourable to E / number of all possible outcomes
• Sum of probabilities of all elementary events = 1
• Always check that the total number of outcomes is counted correctly

Some events can never happen. For example, in a single throw of a die, getting the number 8 is impossible because no face of the die is marked 8. The number of favourable outcomes is 0, so the probability is 0/6 = 0. Such an event is called an impossible event. On the other hand, some events are certain to happen. In a single throw of a die, getting a number less than 7 is sure because every face is marked with a number less than 7. Here the number of favourable outcomes equals the total number of outcomes, so the probability is 6/6 = 1. Such an event is called a sure event or a certain event. Since the number of favourable outcomes is always less than or equal to the total number of outcomes, the probability of any event E always lies between 0 and 1, that is, 0 ≤ P(E) ≤ 1.

• Impossible eventprobability 0
• Sure (certain) eventprobability 1
• For any event E, 0 ≤ P(E) ≤ 1

The event 'not E', denoted by E with a bar over it, is called the complement of the event E. The events E and 'not E' are called complementary events. For example, if E is the event 'getting a head' when a coin is tossed, then 'not E' is the event 'getting a tail'. Since one of E or 'not E' must happen, their probabilities add up to 1: P(E) + P(not E) = 1. This gives the useful formula P(not E) = 1 − P(E). For instance, if the probability that Sangeeta wins a tennis match is 0.62, then the probability that Reshma wins is 1 − 0.62 = 0.38. Similarly, if the probability that a card drawn from a deck is an ace is 1/13, the probability that it is not an ace is 1 − 1/13 = 12/13.

• P(E) + P(not E) = 1
• P(not E) = 1 − P(E)
• E and 'not E' are called complementary events

Probability is used in many everyday and real-life situations. In a class of 40 students with 25 girls and 15 boys, if one name is drawn at random, the probability of drawing a girl's name is 25/40 = 5/8 and of drawing a boy's name is 15/40 = 3/8. In a carton of 100 shirts with 88 good, 8 with minor defects and 4 with major defects, the probability that a shirt drawn at random is acceptable to a trader who accepts only good shirts is 88/100 = 0.88, while for a trader who rejects only majorly defective shirts it is 96/100 = 0.96. Probability also helps in situations like finding the chance that two friends share the same birthday, or that a missing helicopter crashed inside a lake shown on a map. In such cases we count favourable outcomes and total outcomes carefully before dividing.

• Use P(E) = favourable outcomes / total outcomes in word problems
• Check whether the events are equally likely before applying the formula
• Complement rule often simplifies calculations