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Today Puzzle #787

Puzzle No. 787鈥� Tuesday 21 July

Today, Billy has bought himself a new phone, with each phone call made, there’s a 1 in 1000 chance that the phone will breakdown. The phone is expensive so Billy asks his younger brother Sammy to pay half. In return, Billy will give Sammy the phone to keep at some point in the future. Sammy agrees, provided there’s at least a 50/50 chance that it’s still working at the time of the exchange. Billy promises to hand over the phone after having made exactly 500 calls. Is this a good deal for Sammy?

Today’s #PuzzleForToday has been set by Sally Calder, Education Actuary at the Institute and Faculty of Actuaries.

Click here for the answer

Yes.

The chances of the phone breaking down during a call are 1 in 1,000, ie 0.001

So the probability of the phone surviving a single call is 0.999

The probability of surviving two calls is 0.999 x 0.999 = 0.998001

…and surviving five calls is 0.999 x 0.999 x 0.999 x 0.999 x 0.999 = 0.995995

If we repeat this 500 times, ie 0.999 to the power 500, we get 0.606, i.e. there’s a little over a 60% chance that the phone will have survived 500 calls, which is a better chance than 50/50

So yes, Bobby should agree to this.

In fact, Billy will have to make 692 calls before the probability reaches 50/50. 0.999 to the power of 692 is 0.500

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