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Conditional Probability: Making Use of More Information

Conditional Probability: Making Use of More Information . “Let’s Make a Deal Problem”. Let’s Make a Deal Problem. Select A Door-- PRIZE behind one door: # 1 #2 #3 After Your Initial Pick, Host Opens a Door Without the Prize and Gives You the Option to Switch Doors

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Conditional Probability: Making Use of More Information

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  1. Conditional Probability:Making Use of More Information “Let’s Make a Deal Problem”

  2. Let’s Make a Deal Problem • Select A Door-- PRIZE behind one door: # 1 #2 #3 • After Your Initial Pick, Host Opens a Door Without the Prize and Gives You the Option to Switch Doors • Does It Make Sense to Switch Your Pick After Seeing A Door Opened? What are the relevant probabilities?

  3. PROBABILITIES WITH LIMITED INFORMATION • Facing all 3 doors, probability of Selecting Prize = 1/3 for each door P[Prize #1] = P[Prize #2] = P[Prize #3] = 1/3 • Suppose Player Chooses # 3. Probability Host opens either of remaining doors = 1/2 P[Open #1] = P[Open #2] = 1/2 • IF HOST OPENS DOOR #2, SWITCH PICK TO DOOR #1?

  4. CONDITIONAL PROBABILITY • THE GENERAL FORMULA P[A*B] = {P[A] * P[B*A]}/ P[B] P[A*B] = the likelihood of event A if event B happens (read “prob of A given B”) P[A] = likelihood of event A based on original information P[B] = likelihood of event B based on original information P[B*A] = likelihood of event B if event A happens (read “prob of B given A”) -- this is the key

  5. Using Conditional Probability in Let’s Make a Deal • Key Information: Host always opens Prize last • Why important? Provides information on P[B*A] from the conditional probability formula -- “Probability Host Opens a Door given Prize Behind a Door”

  6. Applying the Formula • THE GENERAL FORMULA P[A*B] = {P[A] * P[B*A]}/ P[B] • Example of use for our problem: P[Prize #3* Opens #2] = P[Prize #3]* P[Open #2 * Prize #3] / P[Open #2] P[Prize #3] = 1/3 (prob w/o any additional information) P[Opens 2] = 1/2 (prob w/o any additional information) P[Opens #2 * Prize #3] = ?? (Prob. Host Opens #2 given Prize in #3)

  7. P[Open #1* Prize in #1] = 0 P[Open #1* Prize in #2] =1 P[Open #1* Prize in #3] = 1/2 P[Open #2* Prize in #1] = 1 P[Open #2* Prize in #2] = 0 P[Open #2 * Prize in #3] =1/2 Making Use of the Information

  8. THE SOLUTION • P[Prize in #1 * Open #2] = P[Prize in #1] * P[Open #2 * Prize in #1] / P[Open #2] = (1/3) * (1) / (1/2) = 2/3 • P[Prize in #3 * Open #2 ] = P [Prize in #3] * P[Open #2 * Prize #3] / P[Open #2] = (1/3) * (1/2) / (1/2) = 1/3

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