Introduction to Probability: CBSE Class 9 Maths

Understanding Probability: The Basics

Probability is a way to quantify the chance of something happening. In our daily lives, we are constantly dealing with uncertainty. Will my favourite team win the match? Will I get the top score in the test? Probability gives us a mathematical tool to analyze these situations. For Class 9, we focus on Experimental Probability. This is different from theoretical probability. Instead of just thinking about what should happen (like a coin having a 1/2 chance of landing on heads), we look at what actually happens when we perform an experiment multiple times. For example, if you toss a coin 100 times and it lands on heads 48 times, the experimental probability of getting a head is 48/100. The core idea is to collect data from trials and use that data to estimate the likelihood of a specific outcome.

Key Terms You Must Know

Trial

A trial is an action or experiment which results in one or several outcomes. For example, tossing a coin is a trial.

Outcome

An outcome is a possible result of a trial. When you toss a coin, the possible outcomes are 'Head' or 'Tail'.

Event

An event is a specific outcome or a collection of outcomes from an experiment that we are interested in. For example, the event of 'getting a Head' when a coin is tossed.

Experimental Probability

Also called Empirical Probability, it is calculated based on the results of an actual experiment. It is the ratio of the number of times an event occurred to the total number of trials conducted.

How to Calculate Experimental Probability: The Formula

1. Step 1: Identify the Event (E) — First, clearly define the event for which you want to find the probability. For instance, the event could be 'rolling a 6 on a die'.
2. Step 2: Count the Number of Favourable Trials — Perform the experiment and count how many times your specific event (E) actually happened. This is the 'number of trials in which the event happened'.
3. Step 3: Count the Total Number of Trials — Note down the total number of times the experiment was performed. This is your 'total number of trials'.
4. Step 4: Apply the Probability Formula — The experimental probability of an event E, written as P(E), is calculated using the formula: P(E) = (Number of trials in which the event happened) / (The total number of trials)

Worked Examples: Seeing Probability in Action

• Example 1: Tossing a Coin A coin is tossed 1000 times with the following frequencies: Heads: 455, Tails: 545 Compute the probability for each event. Solution: Total number of trials: 1000 Event 1: Getting a Head (E₁) Number of trials where a head appeared = 455 Probability of getting a head, P(E₁) = (Number of heads) / (Total number of tosses) = 455 / 1000 = 0.455 Event 2: Getting a Tail (E₂) Number of trials where a tail appeared = 545 Probability of getting a tail, P(E₂) = (Number of tails) / (Total number of tosses) = 545 / 1000 = 0.545 Check: Notice that P(E₁) + P(E₂) = 0.455 + 0.545 = 1.000. The sum of probabilities of all possible outcomes is 1.
• Example 2: Rolling a Die A die is thrown 500 times. The frequencies of the outcomes 1, 2, 3, 4, 5, and 6 are given in the following table: Outcome: 1, 2, 3, 4, 5, 6 Frequency: 90, 75, 80, 65, 95, 95 Find the probability of getting a '3'. Solution: Total number of trials: 500 Event (E): Getting a '3' Number of times '3' appeared (frequency) = 80 Calculate the probability: P(getting a 3) = (Frequency of 3) / (Total number of throws) = 80 / 500 Simplifying the fraction: P(getting a 3) = 8 / 50 = 4 / 25 = 0.16

Key Properties and Common Mistakes

Important Properties of Probability:

1. Probability is always between 0 and 1: The probability of any event E will always be a number from 0 to 1, inclusive. This can be written as 0 ≤ P(E) ≤ 1. If P(E) = 0, it is an impossible event. If P(E) = 1, it is a sure or certain event.
2. Sum of Probabilities: The sum of the probabilities of all the possible elementary outcomes of an experiment is always 1.

Common Mistake to Avoid:
A very common error is to reverse the formula. Students sometimes write (Total number of trials) / (Number of favourable trials). This is incorrect and will give you a number greater than 1 (unless the event happened in every trial). Always remember: Part / Whole. The number of times your event happened is the 'part', and the total number of trials is the 'whole'.

Practice Questions with Solutions

• Q: Two coins are tossed simultaneously 500 times and we get: Two heads: 105 times One head: 275 times No head: 120 times Find the probability of occurrence of each of these events. A: Step 1: Identify the total number of trials. The coins were tossed 500 times, so the total number of trials is 500. Step 2: Calculate the probability for 'Two heads' (Event E1). The number of times two heads appeared is 105. P(E1) = 105 / 500 = 21 / 100 = 0.21. Step 3: Calculate the probability for 'One head' (Event E2). The number of times one head appeared is 275. P(E2) = 275 / 500 = 11 / 20 = 0.55. Step 4: Calculate the probability for 'No head' (Event E3). The number of times no head appeared is 120. P(E3) = 120 / 500 = 12 / 50 = 6 / 25 = 0.24. Final answer: P(Two heads) = 0.21, P(One head) = 0.55, P(No head) = 0.24. (Check: 0.21 + 0.55 + 0.24 = 1.00)
• Q: A survey of 200 students was conducted about their opinion on a new school rule. 135 students were in favour of the rule, while 65 were against it. Find the probability that a student chosen at random is (i) in favour of the rule, (ii) against the rule. A: Step 1: Identify the total number of students surveyed, which is the total number of trials. Total trials = 200. Step 2: Calculate the probability for a student being 'in favour'. Number of favourable outcomes = 135. P(in favour) = 135 / 200 = 27 / 40 = 0.675. Step 3: Calculate the probability for a student being 'against'. Number of favourable outcomes = 65. P(against) = 65 / 200 = 13 / 40 = 0.325. Final answer: The probability of a student being in favour is 0.675, and the probability of being against is 0.325.
• Q: A student's marks in 5 monthly unit tests are 78, 85, 92, 65, and 80 out of 100. Find the probability that in the next test, the student scores more than 80 marks. A: Step 1: Identify the total number of trials. There have been 5 tests, so the total number of trials is 5. Step 2: Define the event. The event is 'scoring more than 80 marks'. Step 3: Count the number of times this event occurred. The scores are 78, 85, 92, 65, 80. The scores more than 80 are 85 and 92. So, the event occurred 2 times. Step 4: Calculate the probability. P(scoring > 80) = (Number of tests with score > 80) / (Total number of tests) = 2 / 5 = 0.4. Final answer: The probability that the student scores more than 80 in the next test is 2/5 or 0.4.
• Q: A bag contains 5 red balls, 8 blue balls, and 7 green balls. A ball is drawn from the bag 100 times, with replacement. The results are: Red: 22, Blue: 45, Green: 33. What is the experimental probability of drawing a blue ball? A: Step 1: Identify the total number of trials. The ball was drawn 100 times, so the total number of trials is 100. Note that the number of balls in the bag (5, 8, 7) is extra information for an experimental probability question. Step 2: Define the event. The event is 'drawing a blue ball'. Step 3: Count the number of times a blue ball was drawn. The result shows that a blue ball was drawn 45 times. Step 4: Calculate the experimental probability. P(blue ball) = (Number of times blue was drawn) / (Total number of draws) = 45 / 100 = 9 / 20 = 0.45. Final answer: The experimental probability of drawing a blue ball is 0.45.

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