Circus of Luck 2

Probability Level 3

Larry enters a newly opened circus which houses 11 11 tents (Tent A , B , C , . . . , K A, B, C, ... , K ) that each features a free-to-play game. The game of Tent A A involves a computer that will flash one from 2 2 symbols. If the contestant guesses what symbol will be flashed, he wins 2 $ 2\$ . The game of Tent B B involves a computer that will flash one from 3 3 symbols. If the contestant guesses what symbol will be flashed, he wins 3 $ 3\$ . The game of Tent C C involves a computer that will flash one from 4 4 symbols. If the contestant guesses what symbol will be flashed, he wins 4 $ 4\$ . And so on. Tent A ( 1 2 chance of winning , 2 $ prize ) , Tent B ( 1 3 chance of winning , 3 $ prize ) , Tent C ( 1 4 chance of winning , 4 $ prize ) , . . . , Tent K ( 1 12 chance of winning , 12 $ prize ) \text{Tent} \space A \space (\dfrac{1}{2} \space \text{chance of winning}, \space 2\$ \space \text{prize}), \space \text{Tent} \space B \space (\dfrac{1}{3} \space \text{chance of winning}, \space 3\$ \space \text{prize}), \space \text{Tent} \space C \space (\dfrac{1}{4} \space \text{chance of winning}, \space 4\$ \space \text{prize}), \space ... \space , \space \text{Tent} \space K \space (\dfrac{1}{12} \space \text{chance of winning}, \space 12\$ \space \text{prize})

Question : If Larry plays all games once each, which of the statements below would be true?

Try Circus of Luck 1 or 3

Winning a total of 77 $ 77\$ is more likely than winning a total of 75 $ 75\$ . Winning a total of 75 $ 75\$ is more likely than winning a total of 77 $ 77\$ . The chance of winning a total of 75 $ 75\$ and winning a total of 77 $ 77\$ is the same.

Kaizen Cyrus by Kaizen Cyrus , 18, Philippines

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3 solutions

Kaizen Cyrus
May 19, 2019

For Larry to win a total of 77 $ 77\$ , he must win all of the games. To win a total of 75 $ 75\$ , he must win all except for Tent A A .

The chance of winning a total of 77 $ 77\$ 1 2 × 1 3 × 1 4 × 1 5 × 1 6 × 1 7 × 1 8 × 1 9 × 1 10 × 1 11 × 1 12 = 2.0877 % \scriptsize \textcolor{#3D99F6}{\dfrac{1}{2}} × \textcolor{#3D99F6}{\dfrac{1}{3}} × \textcolor{#3D99F6}{\dfrac{1}{4}} × \textcolor{#3D99F6}{\dfrac{1}{5}} × \textcolor{#3D99F6}{\dfrac{1}{6}} × \textcolor{#3D99F6}{\dfrac{1}{7}} × \textcolor{#3D99F6}{\dfrac{1}{8}} × \textcolor{#3D99F6}{\dfrac{1}{9}} × \textcolor{#3D99F6}{\dfrac{1}{10}} × \textcolor{#3D99F6}{\dfrac{1}{11}} × \textcolor{#3D99F6}{\dfrac{1}{12}} = 2.0877\%

The chance of winning a total of 75 $ 75\$ 1 2 × 1 3 × 1 4 × 1 5 × 1 6 × 1 7 × 1 8 × 1 9 × 1 10 × 1 11 × 1 12 = 2.0877 % \scriptsize \textcolor{#D61F06}{\dfrac{1}{2}} × \textcolor{#3D99F6}{\dfrac{1}{3}} × \textcolor{#3D99F6}{\dfrac{1}{4}} × \textcolor{#3D99F6}{\dfrac{1}{5}} × \textcolor{#3D99F6}{\dfrac{1}{6}} × \textcolor{#3D99F6}{\dfrac{1}{7}} × \textcolor{#3D99F6}{\dfrac{1}{8}} × \textcolor{#3D99F6}{\dfrac{1}{9}} × \textcolor{#3D99F6}{\dfrac{1}{10}} × \textcolor{#3D99F6}{\dfrac{1}{11}} × \textcolor{#3D99F6}{\dfrac{1}{12}} = 2.0877\%

Since the chance of winning Tent A A is as likely as losing on it, the chance of winning a total of 75 $ 75\$ and winning a total of 77 $ 77\$ is the same.

0 Helpful 0 Interesting 2 Brilliant 0 Confused
Elijah Frank
Dec 6, 2020

the difference of winning rate between 77$ and 75$ is 1/2. Curious fact: 1/2 of people do correct in this problems the same as the ans.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
G Silb
May 20, 2019

$77 is $2 more than $75. The chance of winning $2 is the same as not winning it: 1/2. Therefore the chances of winning $77 and $75 are the same.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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