Conditional Probability

John Turner revision date Sept 5, 1996.

Return to syllabus.

These are supplemental problems for Chapter 4 of SM230. They were written by J.C. Turner.

  1. A certain carrier pilot catches the number three wire on 60% of his landings. If he lands three times, assumed independent, on a given day, what is the probability that he will catch the number three wire (a) all three times; (b) at least once; (c) at least twice.
  2. Show that for any events A and B that the probability that exactly one of the events occurs is P(A) + P(B) - 2P(A&B), where A&B is the intersection.
  3. A manufacturer of magnetrons has 3 plants. 30% of the magnetrons come from plant A, 50% from B, and 20% from C. Based on past history, 1 out of every 100 tubes from plant A is defective, 1 out of 200 from B, and 1 out of 50 from plant C.
    1. What is the probability that a magnetron you draw from supply is defective?
    2. Given that it is defective, what is the probability it came from plant A? B? C?
  4. You are planning both air and cruise missile strikes against a desalinization plant in the Arabian Gulf. You determine that the probability that an air strike is successful is 1/3. The probability that a cruise missile strike is successful is 2/3. The strikes are independent of each other.
    1. What is the probability that both strikes are successful?
    2. What is the probability that only the airstrike is successful?
    3. What is the probability that the target is successfully
    4. attacked (by at least one strike)?
    5. Now suppose that the probability of a successful airstrike is not independent of the probability of a successful cruise missile attack, but that if the missile attack is successful, then there is a 75% chance that the air strike will be successful. What is the probability that the missile attack is successful and the airstrike is unsuccessful?
  5. Two newly commissioned ensigns, nicknamed Flounder and Viper, are roommates at Nuke Power School. If the probability that Flounder passes is 0.6, the probability that Viper passes is 0.5, and the probability that they both pass is 0.2.
    1. What is the probability that Viper passes if Flounder passes?
    2. What is the probability the Flounder passes if Viper passes?
    3. Is the probability that they pass independent? Why or why not?
  6. You are the TACCO on a P3. The probability that the buoy field of the P3 acquires a given target is 0.6, the probability that the destroyer acquires it is 0.4, and the probability both acquire it is 0.2.
    1. What is the probability that the P3 acquires the target if the destroyer has contact?
    2. What is the probability that the destroyer acquires contact if the P3 has contact?
    3. Is the probability that the P3 acquires contact independent of the event that the destroyer acquires contact? Why or why not?
  7. Our aircraft has both a primary and backup COM (communication) system. There is an 8% chance of primary failure, and 11% chance that the backup will fail. If the primary fails, there is a 20% chance that the backup will also fail.
    1. What is the chance of a complete COM failure (primary and backup)?
    2. Is backup failure independent of primary? Why?
    3. If they were independent, how often would both systems fail?
  8. The attack is set for dawn. If we have complete surprise, we are sure to win. If we have only partial surprise, we have a 50% chance of winning. If we have no surprise, we have only a 10% chance of winning. There is a 60% chance of complete surprise and a 25% chance of partial surprise (the rest of the time there is no surprise).
    1. What is the chance of winning?
    2. If we have won, what is the chance it was without benefit of surprise?
  9. Shirts come in 3 sizes, Small, Medium, Large. They also come in 3 collar styles, Button Down (BD), Tab and Fly. 35% of shirts are Small and 25% are Large. 40% are BD and 35% are Tab.
    1. How many are Medium?
    2. How many are Fly?
    3. Suppose 12% are Small and BD, 14% are Small and Tab, 10% are Medium and Tab, 2% are Large and Fly.
    4. For all other combinations of size and collar, how many shirts fall into each category?
    5. Given the shirt size, what is the conditional probability of each collar style?
    6. Given collar style, what is the conditional probability of each size?
  10. In what follows, U means "union", & means "intersection" and c means complement, as in Ac. Suppose P(A)=0.73 and P(B)=0.52.
    1. What is the minimum value for P(A&B)?
    2. If P(A&B)=0.37, what is P(A U B)?
    3. Assuming (b), what is P((A&B)c)? (The complement of A&B.)
    4. What is P(Ac & B)?
  11. Suppose 81% of enemy subs have acoustic tiles, 34% have active degaussing and 23% have neither. How many have one or the other, but not both?
  12. Suppose 23% of enemy aircraft have the new insignia, 35% have updated engines and 10% have both.
    1. How many have either?
    2. How many have one, but not the other?
  13. 60% of the seafood in area stores comes from the East Coast, 25% comes from the Gulf Coast and the rest (15%) comes from the West Coast. 40% of the seafood is fresh and from the East Coast, 10% is fresh from the Gulf Coast and 4% is fresh from the West Coast.
    1. What fraction of (total) fish is fresh?
    2. How much frozen fish comes from the Gulf Coast?
    3. How much fresh fish comes from either the East or Gulf Coasts?
John Turner
jct@usna.navy.mil

Return to syllabus.