CAT Questions | CAT Algebra Questions: Probability Problems

• Experiment: It is an operation that can produce some well-defined outcomes
• Random Experiment: If each of the trials of an experiment is conducted under identical conditions, then the outcome is not unique, but maybe of any possible outcome then such an experiment is known as a random experiment.
• Sample Space: The set of all possible outcomes in a Random Experiment is known as Sample space, provided no two or more of these outcomes can occur simultaneously and exactly one of these outcomes must occur whenever the experiment is conducted.
• Event: Any subset of a sample space is called an event.
• Certain and Impossible events: If S is a sample space, then both S and null set φ both are events. S is called a certain event and φ is called an impossible event.
• Equally Likely Events: The given events are said to be equally likely if none of them is expected to occur in preference of the other.
• Exhaustive events: In probability theory, s system of events is called exhaustive, if at least one of the event of the system occurs. Ex. If a coin is tossed then Head and Tails form an exhaustive set of events.
• Mutually Exclusive events: A set of events is called mutually exclusive events if the occurrence of one of them precludes the occurrence of any of the remaining events. In other words, if there are two events P and B then they are called mutually exclusive if PꓵB = φ.
• Independent events: P and Q are two events. Then,

1. The event ‘either P or Q’ is said to occur if at least one of P and Q occurs. It is usually denoted as P ꓴ Q (P or Q).
2. The event ‘both P and Q’ is said to occur if both P as well Q occur simultaneously. It is represented as P ꓵ Q (P and Q).
3. The event ‘P not’ occurs if P does not occur. It is written as Pᶜ.
4. The event ‘P but not Q’ is said to occur if P occurs but Q does not occur. It is denoted by P – Q.

• Probability of an event: It means in the performance of a random experiment the occurrence of any event is always uncertain but a measure of its probable occurrence can be devised known as the probability of the event.

1. The probability of the null event is 0.
2. The probability of a sure event is 1.
3. 0 ≤ P(E)≤1
4. ∑P(E)= 1

• Let S denotes the sample space of a random experiment and P be an event. Then the probability of P is defined as No. of favorable outcomes for event P

Total no. of outcomes. =n(P)/ n(S).

• Some important theorems

1. If P is a subset of Q then, P(P)≤P(Q).
2. P(φ) = 0.
3. P(S) = 1.
4. P(Pᶜ) = 1 – P(P).
5. P(Q-P) = P[Q-(PꓵQ)] = P(Q)-P(PꓵQ).
6. P(PꓴQ) = P(P) + P(Q) – P(PꓵQ).
7. P(PꓴQ) = P(P) + P(Q) when P(PꓵQ) = φ.
8. P(PꓴQꓴR) = P(P) + P(Q) + P(R) – P(PꓵQ) – P(QꓵR) – P(RꓵP) + P(PꓵQꓵR).

• Conditional Probability:

Let S be the sample space. Let P and Q be any two events. P ≠ φ. Then, the probability of event Q, if P has already occurred, is called the conditional probability of Q restricted to the occurrence of P. It is represented as P(P/Q). Thus, the probability of the event Q restricted to the occurrence of event P is the same as the probability of event PꓵQ while P is considered as sample space.

P(Q/P) = n(PꓵQ)/ n(P) = P(PꓵQ)/P(P)

P(PꓵQ) = P(P). P(Q/P)

If P ≠ φ & Q ≠ φ then,

P(PꓵQ) = P(P). P(Q/P) = P(P). P(P/Q).

• Conditional probability is a very important concept of probability. It is used in a variety of questions. Let’s ingrain the concept more comprehensively using the example given below.

Independent Events: Two events are said to be independent if the probable occurrence or non-occurrence of anyone is not affected by the occurrence or non-occurrence of the other i.e. two events P and Q are independent if

• P(P/Q) = P(P/Qᶜ) = P(P)
• Or, P(Q/P) = P(Q/Pᶜ) = P(Q)
• Or, P(PꓵQ) = P(P). P(Q)
  The relation between Independent and Mutually Exclusiveness of two events.
• If two events P ≠ φ and Q ≠ φ are independent, then they are not mutually exclusive.
• If two events P ≠ φ and Q ≠ φ are mutually exclusive, then they are not independent.
• Independent Experiments:  Let there be two random experiments one after the other. If on repeated performances the sample space of any is not affected by the result of the other, then two experiments are independent of each other.
• There’s a difference between independent events and independent experiments, but students tend to confuse between the two. The former talks about events in an experiment and the latter explain independence among multiple experiments.
• Bayes’ Theorem: Suppose P₁, P₂, … Pn, are a mutually exclusive and exhaustive set of events. Thus, they divide the sample into n parts and event Q occurs. Then the conditional probability that Pi happen given that Q has happened is given by

P(P/Q) = P(Pi). P(Q/Pi)

∑ni=1 P(Pi). P(Q/Pi)