(i) The classical approach to probability theory requires that the total number of possible outcomes be known or calculated and that each of the outcomes be equally likely.
(ii) A marginal probability is also known as unconditional probability.
(iii) For three independent events, the joint probability of the three events, $P(ABC) = P(A) \times P(B/A) \times P(C/AB)$
(iv) Two events are mutually exclusive, exhaustive and equally likely, the probability of either event A or B or both occurring $P(A \text{ or } B) = P(A) + P(B)$
Solution Analysis
We evaluate each statement concerning probability theory to determine its accuracy.
Statement (i): Classical Probability Requirements
The classical approach to probability assumes a finite set of possible outcomes where each outcome is equally likely. Calculating probability requires knowing the total number of possible outcomes and the number of outcomes favorable to the event.
Verdict: True
Statement (ii): Marginal vs. Unconditional Probability
Marginal probability is the probability of a single event occurring, denoted as $P(A)$. This is equivalent to the definition of unconditional probability, as it does not depend on the outcome of any other event.
Verdict: True
Statement (iii): Joint Probability for Independent Events
The formula presented, $P(ABC) = P(A) \times P(B/A) \times P(C/AB)$, is the general multiplication rule for joint probability.
For three events to be independent, the probability of their intersection is simply the product of their individual probabilities: $P(ABC) = P(A) \times P(B) \times P(C)$. The conditional probabilities $P(B/A)$ and $P(C/AB)$ would equal $P(B)$ and $P(C)$ respectively if the events were independent.
The statement incorrectly applies the general multiplication rule as the rule for independent events.
Verdict: False
Statement (iv): Probability of Union for Mutually Exclusive Events
The formula $P(A \text{ or } B) = P(A) + P(B)$ is the addition rule for probabilities. This rule is valid when events A and B are mutually exclusive, meaning they cannot happen at the same time ($P(A \text{ and } B) = 0$).
The statement specifies that the events are mutually exclusive (along with exhaustive and equally likely). The condition of mutual exclusivity is sufficient for the formula $P(A \text{ or } B) = P(A) + P(B)$ to hold true.
Verdict: True
Conclusion
Statements (i), (ii), and (iv) are accurate descriptions within probability theory. Statement (iii) presents an incorrect formula for independent events.
Therefore, the sentences (i), (ii), and (iv) are true.
Important Questions from Basics of Probability
Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ______ . (rounded off to three decimal places)
- A quadratic polynomial $(x - \alpha)(x - \beta)$ over complex numbers is said to be square invariant if $(x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random.
The probability that the roots of the chosen polynomial are equal is ____________ (rounded off to one decimal place) - If, out of 50 fish of a pond, 12 had no ectoparasite and remaining had varying numbers of parasites on them, then how many of fish had at least one parasite ?
For a distribution with mean, median, mode and standard deviation 25, 24, 26 and 5 respectively, Karl Pearson's coefficient of skewness equals to:
Suppose two distributions have the same mean, same standard deviation and are equally skewed, but the first distribution is more peaked than the
other. This feature of the frequency distributions in which they are different are called:
