Probability of the intersection of two events
Table of Content:
Formula
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Probability of the intersection of two independent events: $$P(A \cap B) = P(A) \cdot P(B)$$
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Probability of the intersection of two dependent events: $$P(A \cap B) = P(A) \cdot P(B | A)$$
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Probability of the intersection of two mutually exclusive events: $$P(A \cap B) = 0$$
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Probability of the intersection of two complementary events: $$P(A \cap A^c) = 0$$
Probability of the intersection of two independent events
Suppose we have two independent events A and B. Then the probability of the intersection of A and B is:
$$P(A \cap B) = P(A) \cdot P(B)$$
For example, if we roll a fair six-sided die, the probability of getting a 4 on the first roll is \(P(A) = \frac{1}{6}\), and the probability of getting a 2 on the second roll is \(P(B) = \frac{1}{6}\). Since the events of getting a 4 on the first roll and getting a 2 on the second roll are independent, the probability of getting a 4 on the first roll and a 2 on the second roll is:
$$P(A \cap B) = P(\text{getting a 4 on the first roll}) \cdot P(\text{getting a 2 on the second roll}) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$$
Therefore, the probability of getting a 4 on the first roll and a 2 on the second roll is \(\frac{1}{36}\).
Probability of the intersection of two dependent events:
The probability of the intersection of two dependent events \(A\) and \(B\) is given by:
$$P(A \cap B) = P(A) \cdot P(B|A)$$
where \(P(B|A)\) is the conditional probability of \(B\) given \(A\). This formula is used when the occurrence of one event affects the probability of the occurrence of the other event.
Suppose we have two dependent events, A and B, and we know that the probability of A is 0.4 and the probability of B given A has occurred is 0.3. Then the probability of the intersection of A and B can be calculated as:
\begin{aligned} P(A \cap B) &= P(A) \cdot P(B \mid A) \\ &= 0.4 \cdot 0.3 \\ &= 0.12 \end{aligned}So the probability of the intersection of A and B is 0.12.
Probability of the intersection of two mutually exclusive events
The probability of the intersection of two mutually exclusive events is \(0\), since if the events are mutually exclusive, they cannot occur at the same time. The statement is:
$$P(A \cap B) = 0$$
where \(A\) and \(B\) are the two mutually exclusive events.
Suppose we flip a fair coin and roll a fair six-sided die. Let event A be the coin landing heads, and let event B be the die showing a 6. Since these events are mutually exclusive (the coin cannot land both heads and tails, and the die cannot show a 6 and not show a 6), the probability of their intersection is 0:
$$P(A \cap B) = P(\varnothing) = 0$$
Probability of the intersection of two complementary events:
The probability of the intersection of two complementary events is always 0, since complementary events are mutually exclusive. In other words, if two events are complementary, they cannot occur simultaneously. Therefore, their intersection is always an empty set, which has a probability of 0.
Here is the syntax for the probability of the intersection of two complementary events:
\begin{equation} P(A \cap A') = \varnothing = 0 \end{equation}
where \(A\) and \(A'\) are complementary events, and \(\varnothing\) represents the empty set.