Math is a tricky subject to navigate. Though there is no substitute for in-depth knowledge or practice, here is a last-minute revision sheet, addressing common errors that students make.
Probability = number of favourable events/ total number of events
This holds only when all events are equally likely to happen. For instance, the probability of getting a six on a loaded die is not 1/6.
Independent events and mutually exclusive events are different. By definition, mutually exclusive events cannot be independent
Making a tree diagram or listing down the favourable events is easier than using formulae in certain cases, such as finding the number of the rolls of two die in which the numbers on the die add up to 4.
Events that appear to be independent at face value may not always be so, and vice versa. It is better to confirm using the multiplication theorem.
Most “at least” questions are easier to handle by finding the probability of their complements and subtracting it from one.
Eg: Find the probability of getting at least one head in five tosses of a coin.
Instead of adding the individual probabilities, i.e., the probabilities of getting one, two, three, four and five heads, it is easier to subtract the probability of getting no heads from 1.
P(at least one head) = 1 – P(no heads)
While solving questions, always define the events that you require.
Choosing certain objects from a given set without replacement in no particular order is equivalent to selecting the same number of objects together from the set. Use combinations in this case to find the number of favourable outcomes.
Eg: The number of ways in which three cards can be selected without replacement in no particular order from a deck of cards without replacement is 52C3.
If the order matters, use conditional probability and the multiplication theorem.
Eg: Three cards are drawn without replacement from a deck of cards. What is the probability that the first two cards are kings and the third is an ace? [NCERT]
Let K be the event of drawing a king and A the event of drawing an ace.
Then P(KKA) = P(K) P(K|K) P(A|KK)
= 4/52 + 3/51 + 4/50
= 2/5525
If combinations are used, you include all three arrangements of the cards, i.e., KKA, KAK and AKK. The probability of obtaining this becomes three times the above probability, which is clearly incorrect.
There may be questions on the theorem of total probability; do remember the formula.
P(A) = P(A|E1) + P(A|E2) + P(A|E3)
It forms the denominator of the formula given by Bayes’ theorem.
While using the Theorem of Total Probability and Bayes’ Theorem:
Ensure that all partitions are mutually exclusive
Ensure that they are mutually exhaustive – no possible case should be left out
Identifying the events correctly: The ‘final’ event – the one which follows a particular event – is taken as event A. The which precede A (often the selection of a box or a bag) would be E1, E2, E3, etc.
Eg: A man is known to speak the truth three out of four times. He throws a die and reports that it is six. Find the probability that it is actually a six. [NCERT]
The man rolls the die first and then makes his statement. Thus the outcomes of roll – getting a six and not getting a six – become events E1and E2 respectively. The final event, i.e., the man reporting that he got a six, is event A.
Random variables and probability distribution: Define what the random variable X is. For example, it is the number of heads in ten tosses of a coin. Verify your answers by ensuring that all Pi’s add up to 1.
Prerequisites for applying the binomial formula for Bernoulli trials
Finite number of trials
Each trial has only two outcomes
Trials are independent
Probability of each outcome remains the same in each trial
For instance, you cannot use the formula to calculate the probability of drawing three black balls without replacement from a bag containing seven black and three white balls. As the balls are not replaced, each draw depends on the preceding one and the probability of drawing a black ball changes each time.
State your answer in the following manner for a question on binomial distribution:
Say you want to answer the question: Find the probability distribution of the number of heads in six tosses of a coin.
1. Define X (the number of heads in six tosses of a coin, in this case).
2. State the values X can take (1,2,3,4,5,6).
3. State that X follows the binomial distribution B (n,p), where n = the number of trials (six) and p = probability of success (probability of getting a head = 1/2)
4. State q = probability of failure (probability of getting tails)
q = 1 – p (=1/2)
5. Construct the probability distribution table
We hope this helped you! Good luck and studying!
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