Question

Laura is super excited to try her new transporter that allows her to teleport through space towards Jupiter, 1 Astronomical Unit (AU) at a time. However, since Laura cannot code, her transporter doesn’t work perfectly. She discovers that each time she presses her TELEPORT button, her Space Transporter 3000 will teleport her 1 AU directly towards Jupiter with probability p, and it will fail, meaning she will not move at all, with probability (1 − p).

(a) What is the probability that after pressing the TELEPORT button twice, Laura will be 1 AU closer to Jupiter than she was when she started?

(b) What is the probability that after pressing the TELEPORT button three times, Laura will be 2 AU closer to Jupiter than she was when she started?

(c) Given that after pressing the TELEPORT button three times, Laura has moved 2 AU closer to Jupiter, what is the probability that after pressing the button the first time, Laura teleported 1 AU closer to Jupiter?

Answer 1

Given, probability of going towards Jupiter by 1 AU = p

probability of not going towards Jupiter by 1 AU = (1 - p)

(a). There are two conditions when she will be closer to Jupiter by 1 AU if TELEPORT button is pressed two times
Case 1: First hit takes 1 AU closer, Second hit doesn't

Case 2: First hit doesn't, Second hit takes 1 AU closer

Hence, P = p*(1-p) + (1-p)*p = 2p(1-p)

(b). There are three conditions when she will be closer to Jupiter by 2 AU if TELEPORT button is pressed three times
Case 1: First hit takes 1 AU closer, Second hit takes 1 AU closer, Third hit doesn't

Case 2: First hit takes 1 AU closer, Second hit doesn't, Third hit takes 1 AU closer

Case 3: First hit hit doesn't, Second takes 1 AU closer, Third hit takes 1 AU closer

Hence, P = p*p*(1-p) + p*(1-p)*p + (1-p)*p*p = 3p²(1-p)

(c). Following from the previous question, the probability Laura is closer by 2 AU, with first hit taking 1 AU closer, leaves us with two cases:

Case 1: First hit takes 1 AU closer, Second hit takes 1 AU closer, Third hit doesn't

Case 2: First hit takes 1 AU closer, Second hit doesn't, Third hit takes 1 AU closer

Probability = p*p*(1-p) + p*(1-p)*p = 2p²(1-p)