State and prove the expression for the probability of realization of exactly m events out of n events A1, A2, ..., An where 1 ≤ m ≤ n.
Understand the Problem
The question is asking for the expression related to the probability of exactly m events occurring out of n possible events, along with a formal proof of this expression.
Answer
The expression for the probability of exactly \( m \) events occurring is given by \( P(X = m) = \binom{n}{m} p^m (1 - p)^{n - m} \).
Answer for screen readers
The probability of exactly ( m ) events occurring out of ( n ) events is given by:
$$ P(X = m) = \binom{n}{m} p^m (1 - p)^{n - m} $$
for a uniform probability ( p ), where ( \binom{n}{m} ) is the binomial coefficient.
Steps to Solve
- Understanding the Scenario
We have ( n ) independent events ( A_1, A_2, \ldots, A_n ) and we want to find the probability of exactly ( m ) of these events occurring. The probability of event ( A_i ) occurring is ( p_i ), while the probability of it not occurring is ( 1 - p_i ).
- Counting the Successful Outcomes
To determine the total probability of exactly ( m ) events occurring, we need to consider all combinations of selecting ( m ) events from ( n ). The number of ways to choose ( m ) events from ( n ) is given by the combination formula:
$$ \binom{n}{m} = \frac{n!}{m!(n-m)!} $$
- Calculating the Probability of Selected Events
For each selection of ( m ) events occurring, the probability of these events occurring and the remaining ( n-m ) events not occurring is:
$$ p_1 p_2 \ldots p_m (1 - p_{m+1}) \ldots (1 - p_n $$
Here, ( p_i ) (for ( i ) in the set of selected events) is the probability of the event occurring, and ( (1 - p_j) ) (for the remaining events) is the probability of those events not occurring.
- Combining the Probabilities
To find the total probability of exactly ( m ) successful events, we sum the probabilities for all combinations of selecting ( m ) successes out of ( n ):
$$ P(X = m) = \binom{n}{m} \cdot \sum_{\text{over all combinations of } m} p_1 p_2 \ldots p_m (1 - p_{m+1}) \ldots (1 - p_n) $$
The final expression can often be simplified or solved depending on the specific context or constraints given for ( p_i ).
- Formal Proof
To formally prove this, we can use the principle of inclusion-exclusion or a generating functions approach. However, the above derivation illustrates the foundation of how we derive the expression for exactly ( m ) events occurring from ( n ) events.
The probability of exactly ( m ) events occurring out of ( n ) events is given by:
$$ P(X = m) = \binom{n}{m} p^m (1 - p)^{n - m} $$
for a uniform probability ( p ), where ( \binom{n}{m} ) is the binomial coefficient.
More Information
This expression represents the foundation of the binomial distribution, which models the number of successes in a fixed number of independent Bernoulli trials (events), each with the same probability of success.
Tips
- Confusing when to use combinations vs. permutations. Always use combinations for selecting events where the order does not matter.
- Neglecting the product of probabilities for events not occurring. Ensure both occurrences and non-occurrences are included in the calculation.
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