Tina randomly selects two distinct
numbers from the set {1, 2, 3, 4, 5}, and
Sergio randomly selects a number from the set {1, 2, . . . , 10}. The probability that Sergio’s number is larger than the sum of the two numbers chosen by Tina is ?
To find the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina, we need to consider all possible outcomes and determine how many of them satisfy this condition.
First, let's list all the possible outcomes for Tina selecting two distinct numbers from the set {1, 2, 3, 4, 5}. There are a total of 5 choices for the first number and 4 choices for the second number (since they need to be distinct). So, there are 5 x 4 = 20 possible outcomes for Tina's choices.
Now, let's consider Sergio's choice. He can select any number from the set {1, 2, ..., 10}. Let's analyze each possible sum of Tina's numbers and count how many choices Sergio has that satisfy the condition of being larger than the sum.
If Tina's numbers sum to 3, Sergio's choices of 4, 5, 6, 7, 8, 9, or 10 will satisfy the condition (7 possible choices).
If the sum is 4, Sergio's choices of 5, 6, 7, 8, 9, or 10 will satisfy the condition (6 possible choices).
If the sum is 5, Sergio's choices of 6, 7, 8, 9, or 10 will satisfy the condition (5 possible choices).
If the sum is 6, Sergio's choices of 7, 8, 9, or 10 will satisfy the condition (4 possible choices).
If the sum is 7, Sergio's choices of 8, 9, or 10 will satisfy the condition (3 possible choices).
If the sum is 8, Sergio's choices of 9 or 10 will satisfy the condition (2 possible choices).
If the sum is 9, Sergio's choice of 10 will satisfy the condition (1 possible choice).
Adding up all the possible choices for Sergio's number, we get 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28.
Since there are 20 possible outcomes for Tina's choices and 28 favorable outcomes for Sergio's choices, the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is 28/20 = 7/5 or 1.4.