Suppose a dot is placed at random in a 10 x 10 graph grid in which squares have been numbered from 1 to 100 with no number repeated. Now simulate choosing 100 numbers at random between 1 and 100 inclusive. How many numbers might you expect to match the square with the dot?
one
To calculate the expected number of matches between the randomly chosen numbers and the square with the dot:
1. Assign variables: Let X represent the number of matches.
2. Calculate the probability of a single match: The dot can be placed in any of the 100 squares with equal probability. Therefore, the probability of a single match is 1/100.
3. Multiply the probability of a single match by the total number of trials (100): E(X) = 1/100 * 100.
4. Simplify the equation: E(X) = 1.
Therefore, you can expect to have 1 number match the square with the dot on average.
To find out how many numbers might match the square with the dot, we can calculate the expected value.
In this case, since there are 100 numbers between 1 and 100 inclusive, the probability of choosing a specific number that matches the square with the dot is 1/100.
The expected value of a random variable represents the average value we expect to obtain if we repeat the experiment many times. In this case, the random variable is the number of matches between the chosen numbers and the square with the dot.
To calculate the expected value, we multiply the probability of each possible outcome by the value of that outcome and sum them up. In this case, the value of each outcome is either 0 or 1, depending on whether the chosen number matches the square with the dot.
Let's calculate it step by step:
1. For the first chosen number, the probability of a match is 1/100, so the expected value for the first number is (1/100) * 1 = 1/100.
2. For the second chosen number, the probability of a match is still 1/100, but there is now one number already chosen that we need to avoid. So the expected value for the second number is (99/100) * (1/100) = 99/10000.
3. Continuing this process for all 100 chosen numbers, the expected value for each number is decreasing by one each time. So the expected value for the third number is (98/100) * (1/100) = 98/10000, and so on.
Finally, we sum up all these expected values to get the total expected value:
(1/100) + (99/10000) + (98/10000) + ... + (2/10000) + (1/10000)
To simplify this sum, we can use the formula for the sum of an arithmetic series:
n * (first term + last term) / 2
where n is the number of terms.
In this case, we have 100 terms, the first term is 1/100, and the last term is 1/10000.
So the total expected value is:
100 * [(1/100) + (1/10000)] / 2 = (100/100) * (101/10000) / 2 = 101/200.
Therefore, we can expect the number of matches between the chosen numbers and the square with the dot to be approximately 101/200, which is approximately 0.505.