This question aims to find the P (S) or P (T) of two mutually exclusive events S and T if the probability of P (S) is given.
Two events are called mutually exclusive if they do not occur at the same time or simultaneously. For Example, when we toss a coin, there are two possibilities whether the head will be displayed or the tail will be displayed on its return. It means both head and tail cannot occur at the same time. It is a mutually exclusive event and the probability of these events occurring at the same time becomes zero. There is another name for mutually exclusive events and that is the disjoint event.
The representation of mutually exclusive events is given as:
\[P (A \cap B) = 0\]
The disjoint events have a rule of addition that is only true only one event is occurring at a time and the sum of this event is the probability of occurrence. Assume two events $A$ or $B$ are occurring then their probability is given by:
\[P (A Or B) = P (A) + P (B)\]
\[P (A \cup B) = P (A) + P (B)\]
When two events $A$ and $B$ are not mutually exclusive events then the formula changes to
\[ P (A \cup B) = P (A) + P (B) – P (A \cap B)\]
If we consider that $A$ and $B$ are mutually exclusive events which means the probability of their occurrence at the same time becomes zero. It can be shown as:
\[P (A \cap B) = 0 \]
Expert Answer
The addition rule of probability is as follows:
\[ P (A \cup B) = P (A) + P (B) – P (A \cap B) \]
This rule in terms of S and T can be written as:
\[ P (S \cup T) = P (S) + P (T) – P (S \cap T) \]
Consider the probability of event T is $ P (T) = 10 $.
By putting values:
\[ P (S \cup T) = 20 + 10 – P (S \cap T) \]
\[ P (S \cup T) = 30 – P (S \cap T) \]
According to the definition of mutually exclusive events:
\[ P (S \cap T) = 0 \]
\[ P (S \cup T) = 30 – 0 \]
\[ P (S \cup T) = 30 \]
Numerical Solution
The probability of occurrence of mutually exclusive events is $ P (S \cup T) = 30 $
Example
Consider two mutually exclusive events M and N having P (M) = 23 and P (N) = 20. Find their P (M) or P (N).
\[ P (M \cup N) = 23 + 20 – P (M \cap N) \]
\[ P (M \cup N) = 43 – P (M \cap N) \]
According to the definition of mutually exclusive events:
\[ P (M \cap N) = 0 \]
\[ P (M \cup N) = 43 – 0 \]
\[ P (M \cup N) = 43 \]
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