We are asked to determine if the given sentence is true or false.
If you roll a dice 10 times, you will roll the expected value at least twice.
Before we start thinking if the sentence is false or not, we will calculate the expected value of a dice roll. The probability of rolling each dice value is the same, 16.
Dice Roll
Probability
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
To calculate the expected value, we need to add up all the products of the possible dice rolls and their probabilities.
E(Die Roll)= 1/6* 1+ 1/6* 2+...+1/6* 6=3.5
The expected value of a dice roll is 3.5. Since the numbers on a dice are all integers, it is impossible to roll exactly 3.5. Therefore, if roll you a dice 10 times you will not roll the expected value, not even once. The given statement is false.
Alternative Solution
Without Calculating the Expected Value
On average we should expect to roll a number higher than 1, so the expected value is definitely bigger than 1. However, a situation where someone rolls 1 ten times in a row is possible and can happen.
This is an example of 10 dice rolls where the expected value did not appear even once. Therefore, this situation is a counterexample to the given statement and it proves that the statement is false.
