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## Problem on Probability from SMO, 2012 | Problem 33

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2012 based on Probability.

## Problem – Probability (SMO Entrance)

Two players A and B play rock – paper – scissors continuously until player A wins 2 consecutive games. Suppose each player is equally likely to use each hand – sign in every game . What is the expected number of games they will play?

• 12
• 15
• 16
• 20

### Key Concepts

Probability

Permutation Combination

Challenges and Thrills – Pre – College Mathematics

## Try with Hints

Two players are playing a series of games of Rock – Paper – scissors. There are a total of K games played. Player 1 has a sequence of moves denoted by string A and similarly player 2 has string B. If any player reaches the end of their string, they move back to the start of the string. The task is to count the number of games won by each of the player when exactly K games are being played.

To start with this particular problem let’s set an expectation as k.

If A doesnot win , so the probability will be = $\frac {no of event}{total event} = \frac {2}{3}$

so game will be restarted.

Try to find out the rest of the cases…………..

Continue from the 1st hint :

Again case 2: If A wins at first and then losses so again the probability will be $\frac {1}{3} \times \frac {2}{3}$. again new game will start.

Last case : A wins in consecutive two games the probability will be $\frac {1}{3} \times \frac{1}{3}$

Do the rest of the sum……..

The total number of games that they will play :

K = $\frac {2}{3} \times (K+1) \times \frac {2}{9} (K+2) \times \frac {1}{9} \times 2$

Now solve this equation and at the end E will be 12. [Check it yourself]

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## What is Average ?

In mathematics and statistics, average refers to the sum of a group of values divided by n, where n is the number of values in the group. An average is also known as a mean.

## Try this sum from AMC 10 – 2020

Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome-it reads the same forward and backward. Then 2 hours later , the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this 2 – hour period ?

A) 50 B) 55 C)60 D) 65 E) 70

American Mathematics Competition 10 (AMC 10B), {2020}, {Problem Number 6}

Average

4 out of 10

Mathematics can be fun

## Use some hints

Do you really need any hint ???

Try this out:

In order to get the smallest palindrome greater than 15951 , we need to raise the middle digit. If we were to raise any of the digits after the middle, we would be forced to also raise a digit before the middle to keep it a palindrome, making it unnecessarily larger.

So what can we do here ?

We can raise 9 to the next largest value, 10 , but obviously, that’s not how place value works, so we’re in the 16000 s now . To keep this a palindrome, our number is now 16061.

If you really need the final hint this can be the life saver for this sum :

So Megan drove 16061 – 15951 = 110 miles . Since this happened over 2 hours , she drove at $\frac {110}{2}$ = 55 mph .