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Explain the concept of a complete sample space and provide the complete sample space for the throwing of two dice.
Explain the concept of a complete sample space and provide the complete sample space for the throwing of two dice.
The complete sample space is the set of all possible outcomes of a random experiment. For the throwing of two dice, the complete sample space consists of all 36 possible pairs of numbers that can appear on the two dice, such as (1,1), (1,2), (1,3), ..., (6,6).
Define the events A and B in the context of throwing two dice, and calculate their probabilities.
Define the events A and B in the context of throwing two dice, and calculate their probabilities.
Event A is the event that the sum of the points on the faces shown is odd. Event B is the event of at least one ace (number ‘1’). The probability of event A is $P(A) = \frac{18}{36} = \frac{1}{2}$, and the probability of event B is $P(B) = 1 - \left(\frac{5}{6}\right)^2 = \frac{11}{36}$.
Calculate the probabilities of the events (i) (A∪B), (ii) (A∩B), and (iii) (A B ∩ ).
Calculate the probabilities of the events (i) (A∪B), (ii) (A∩B), and (iii) (A B ∩ ).
(i) The probability of the union of events A and B is $P(A\cup B) = P(A) + P(B) - P(A\cap B) = \frac{1}{2} + \frac{11}{36} - P(A\cap B)$. (ii) The probability of the intersection of events A and B is $P(A\cap B) = P(A) \times P(B) = \frac{1}{2} \times \frac{11}{36}$. (iii) The probability of the complement of event A intersected with event B is $P(A^c \cap B) = P(B) - P(A\cap B)$.
