You forget to set your alarm with a probaility of .3. If you set the alarm it rings with the probabilty of .8. If the alarm rings, it will wake you on time to make your first class with the probability of .9. If the alarm does not ring, you wake in time for your first class with a probability of .2. What is the probabiltity that you wake in time to make your first class tomorrow?
Pr= get to school|didnt ring * pr didn't ring + get to school|did ring * pr did ring
Pr= .8*.9 + .3*.2= .72+.06
To solve this problem, we can use the concept of conditional probability.
Let's break down the problem into steps:
Step 1: Determine the probability that you forget to set your alarm.
- The probability of forgetting to set the alarm is given as 0.3 (P(forget alarm) = 0.3).
Step 2: Determine the probability that you remember to set your alarm.
- The probability of remembering to set the alarm is the complement of forgetting, which is 1 - 0.3 = 0.7 (P(set alarm) = 0.7).
Step 3: Determine the probability that the alarm rings, given that you remembered to set it.
- The probability of the alarm ringing, given that you set it, is given as 0.8 (P(ring | set alarm) = 0.8).
Step 4: Determine the probability that the alarm does not ring, given that you remembered to set it.
- The probability of the alarm not ringing, given that you set it, is given as 1 - 0.8 = 0.2 (P(not ring | set alarm) = 0.2).
Step 5: Determine the probability that you wake in time to make your first class, given that the alarm rings.
- The probability of waking on time, given that the alarm rings, is given as 0.9 (P(wake on time | ring) = 0.9).
Step 6: Determine the probability that you wake in time to make your first class, given that the alarm does not ring.
- The probability of waking on time, given that the alarm does not ring, is given as 0.2 (P(wake on time | not ring) = 0.2).
Step 7: Use the total probability theorem to calculate the probability that you wake in time to make your first class.
- We can calculate this by considering both cases: the alarm ringing and the alarm not ringing.
- P(wake on time) = P(wake on time | ring) * P(ring) + P(wake on time | not ring) * P(not ring)
- P(wake on time) = 0.9 * 0.8 + 0.2 * 0.2 = 0.72 + 0.04 = 0.76
Therefore, the probability that you will wake in time to make your first class tomorrow is 0.76 or 76%.