Haryana State Board HBSE 10th Class Maths Notes Chapter 15 Probability Notes.
Haryana Board 10th Class Maths Notes Chapter 15 Probability
Introduction
In class IX, we have learnt about experimental (or empirical) probabilities of events which were bases on the results of actual experiments.
Suppose we toss a coin 1000 times and get head say, 430 times and tail 570 times. Then we would say that in a single throw of a coin the probability of getting a head is \(\frac{430}{1000}\) i.e, 0.430 and getting a tail is \(\frac{570}{1000}\) i.e, 0.570. These probabilities are based on the results of an actual experiment of tossing a coin 1000 times. For this reason, these are called experimental or empirical probabilities.
Let us recall some of the basic terms and results that we have studied in the class IX along with a few new ones which are normally used in study of probability.
1. Experiment: Any process that yields a result or an observation is called experiment.
2. Trial: Performing an experiment once is called a trial.
3. Outcome: A particular result of an experiment is called an outcome.
4. Sample space: The set of all possible outcomes of an experiment is called sample space. The individual outcomes in a sample space are called sample points.
e.g. one toss of a coin results in the outcomes (H. T). If the coin in fair, then each out come is equally likely. Two tosses of a coin results in the outcomes (H H, H T, T H, T T).
5. Event: Any subset of the sample space is called an event. If A is an event, then n (A) is the number of same points that belongs to A. e.g. consider the experiment of tossing a die here S = (1, 2, 3, 4, 5, 6)
If A is the event that an odd number occurs, then A = {1, 3, 5}
If E is the even number greater than 4 occurs then E = {6}
6. Equally likely events: If one event cannot be expected in preference to other event then they are said to be equally likely.
For example when we throw a die once, each of the numbers (1, 2, 3, 4, 5, 6) has the of showing up. So, 1, 2, 3, 4, 5, 6 are outcomes of throwing a die.
7. Impossible Event: An event which cannot occur is called an impossible event. eg. throwing a die, 7 will never comes up. So, getting 7 is an impossible event. The probability of an impossibe event is zero.
8. Sure Event: An event which is certain to occur is called a sure event eg. in a single throwing of die, the event to get a number less than 7 is a sure event. The probability of a sure event is 1.
9. Complimentary event: If E denotes the happening of an event and ‘not E’ its not happening the event, then E and “not E” are the complimentary events.
∴ P(E) + P(not E) = 1 P(E) = 1 – P (not E)
10. Probability: The probability of an event A, denoted by P(A), is a measure of the possibility of the event occuring as the result of an experiment.
11. Empirical probability: The probability that a fair die will show a four when thrown is \(\frac{1}{6}\), using an argument based on equally likely outcomes.
12. Theoretical probability: It is P(E) of an event is the fraction of times we expect E to occur.
13. Elementary Event: An event having only one outcome is called an elementary event.
14. Random Experiments: The experiments which have not fixed results are called random experiments.
15. Favourable outcome: The possible outcomes for a given event are called favourable outcomes.
16. Die (Dice): A small cube with its faces numbered from 1 to 6. When the die is thrown, the probability that any particular number from 1 to 6 is obtained on the face landing upper most is \(\frac{1}{6}\).
17. At least: As much as.
18. At most: Not more than.
Probability-A Theoretical Approach
In mathematics probability is the numerical value assigned to the likelihood that a particular event will take place. For instance, if we throw an unbiased die, we have equal chances of scoring any of the numbers 1, 2, 3, 4, 5, and 6. Since there is one chance in six of throwing a 3, the probability of the event occuring is said to be \(\frac{1}{6}\). Similarly when tossing a coin the probability that it lands head is consider to be \(\frac{1}{2}\).
Theoretical probability (also called classical probability) of an event E, written as P(E), is defined as
P(E) \(=\frac{\text { Number of outcomes favourable to } \mathrm{E}}{\text { Number of all possible outcomes of the experiment }}\)
Remark: The value of probability of an event cannot be negative or greater than 1.
Some information related to the playing cards
- A deck of playing cards has in all 52 cards.
- 52 cards divided into 4 suits (spades, clubs, hearts and diamonds). Each suit has 13 cards.
- Cards of heart and diamond are red cards.
- Cards of spades and clubs are black cards.
- King, Queen and Jack are called face cards. Thus, there are in all 12 face cards.
- The total number of non face card is 52 – 12 = 40.