Suppose that each day there is a 20% chance of a fire in a certain city. Each day is independent from the other days. What's the percentage chance that there is a fire on Monday or... Suppose that each day there is a 20% chance of a fire in a certain city. Each day is independent from the other days. What's the percentage chance that there is a fire on Monday or Tuesday (that is, that there is a fire on AT LEAST one of those two days)?
Understand the Problem
The question is asking for the probability of at least one fire occurring on Monday or Tuesday, given a 20% chance per day and assuming the days are independent. To solve this, we can use the complementary probability approach to find the chance of no fires and subtract it from 1.
Answer
$36\%$
Answer for screen readers
The percentage chance that there is a fire on Monday or Tuesday is $36%$.
Steps to Solve
- Define the probability of no fire each day
The chance of a fire on Monday or Tuesday is 20%. Therefore, the chance of no fire on either of those days is $1 - 0.2 = 0.8$ or 80%.
- Calculate the probability of no fire on both days
Since the days are independent, the probability of no fire on both days can be calculated by multiplying their individual probabilities: $$ P(\text{no fire on Monday and no fire on Tuesday}) = P(\text{no fire on Monday}) \times P(\text{no fire on Tuesday}) = 0.8 \times 0.8 = 0.64 $$
- Find the probability of at least one fire
To find the probability of at least one fire on Monday or Tuesday, subtract the probability of no fire on both days from 1: $$ P(\text{at least one fire}) = 1 - P(\text{no fire on Monday and no fire on Tuesday}) $$ Substituting the values: $$ P(\text{at least one fire}) = 1 - 0.64 = 0.36 $$
- Convert the probability to percentage
To express this probability as a percentage, multiply by 100: $$ P(\text{at least one fire}) \times 100 = 0.36 \times 100 = 36% $$
The percentage chance that there is a fire on Monday or Tuesday is $36%$.
More Information
This probability calculation uses the complementary probability approach, which is often simpler when dealing with "at least one" type questions. The calculation shows that in a binary situation (fire/no fire) across independent days, the combined outcomes can be systematically derived.
Tips
- Confusing the probabilities: Ensure to properly distinguish between the chance of a fire occurring and the chance of no fire.
- Forgetting independence: Remember that probabilities of independent events multiply, which is crucial in this problem.
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