3 A multiple choice test paper consists of 20 questions. In each question the candidate has to choose the correct answer from a list of four alternative answers. Suppose the candidate answers each question by selecting an alternative at random. (a) Give an expression for the probability function of X, the number of correctly answered questions. What are the mean and standard deviation of X? (b) If each correct answer is awarded 3 marks, and each incorrect answer carries a penalty of 1 mark, determine the mean and standard deviation of the total mark obtained by the candidate. What is the probability that the total mark is more than zero? (c) Suppose now the test paper contains n questions. Consider the probability that the can- didate answers at least one question incorrectly. What is the least number of questions that the test paper would have to contain in order for this probability to exceed 95%.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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