2. Amitesh estimates that he has a $70 \%$ chance of making the basketball team and a $20 \%$ chance of having failed his last geometry quiz. He defines a "really bad day" as one in which he gets cut from the team and fails his quiz. Assuming that Amitesh will receive both pieces of news tomorrow, how likely is it that he will have a really bad day?

Step 1 ：Amitesh estimates that he has a $70 \%$ chance of making the basketball team and a $20 \%$ chance of failing his last geometry quiz. He defines a 'really bad day' as one in which he gets cut from the team and fails his quiz. Assuming that Amitesh will receive both pieces of news tomorrow, we are asked to find out how likely it is that he will have a really bad day.

Step 2 ：The probability of two independent events both happening is the product of their individual probabilities. In this case, the probability of Amitesh not making the basketball team is $1 - 0.70 = 0.30$ or $30 \%$, and the probability of him failing his geometry quiz is $0.20$ or $20 \%$.

Step 3 ：So, the probability of both events happening is $0.30 \times 0.20 = 0.06$ or $6 \%$.

Step 4 ：The probability of Amitesh having a really bad day, as defined by him, is approximately 0.06 or 6%. This is the product of the probability of him not making the basketball team and the probability of him failing his geometry quiz.

Step 5 ：Final Answer: The probability that Amitesh will have a really bad day is \(\boxed{6 \%}\).