Suppose that a number x is to be selected from the real line R, and let A, B, and C be the events represented by the following subsets of R, where the notation {x : −−−} denotes the set containing every point x for which the property presented following the colon is satisfied: A = {x : 1 ≤ x ≤ 5} B = {x : 3 < x ≤ 7} C = {x : x ≤ 0} Describe each of the following events as a set of real numbers: (a) Ac (b) A ∪ B (c) B ∩ C c (d) (A ∪ B) ∩ C
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.
Suppose that a number x is to be selected from the real line R, and let A, B, and C be the events
represented by the following subsets of R, where the notation {x : −−−} denotes the set containing
every point x for which the property presented following the colon is satisfied:
A = {x : 1 ≤ x ≤ 5}
B = {x : 3 < x ≤ 7}
C = {x : x ≤ 0}
Describe each of the following events as a set of real numbers:
(a) Ac
(b) A ∪ B
(c) B ∩ C c
(d) (A ∪ B) ∩ C
Problem 1.2
Toss a coin 4 times. Let A denote the
denote the event that a head is obtained on the fourth toss. Is A ∩ B empty?
Problem 1.3
Consider rolling a six-sided die once. Let A be the set of outcomes where an odd number comes up.
Let B be the set of outcomes where a 1 or a 2 comes up. Calculate the sets on both sides of the
equalities (Ac ∩ Bc)
c = A ∪ B and (Ac ∪ Bc)
c = A ∩ B and verify that the equalities hold.
Problem 1.4
We are given that P(A) = 0.55, P(B) = 0.35, and P(A∪B) = 0.75. Determine P(Ac) and P(A∩B).
Problem 1.5
Out of the students in a class, 60% are geniuses, 70% love chocolate and 40% fall into both categories. Determine the probability that a randomly selected student is neither a genius nor a chocolate lover.
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