7.5 Basic Concepts of Probability

Consider an experiment that consists of rolling a single standard 6-sided die (with faces numbered 1-6). Decide if these probabilities are equal to zero, equal to one, or somewhere in between.

1. $P \left(\text{roll a 4}\right)$
2. $P \left(\text{roll a 7}\right)$
3. $P \left(\text{roll a positive number}\right)$
4. $P \left(\text{roll a} \frac{1}{3}\right)$
5. $P \left(\text{roll an even number}\right)$
6. $P \left(\text{roll a single}-\text{digit number}\right)$

Recall that a standard deck of cards consists of 52 unique cards which are labeled with a rank (the whole numbers from 2 to 10, plus J, Q, K, and A) and a suit ($♣$, $♢$, $♡$, or $♠$). A standard deck is thoroughly shuffled, and you draw one card at random (so every card has an equal chance of being drawn). Find the theoretical probability of each of these events:

1. The card is $10 ♠$.
2. The card is a $♡$.
3. The…

Assign an empirical probability to the following events:

1. Jose is on the basketball court practicing his shots from the free throw line. He made 47 out of his last 80 attempts. What is the probability he makes his next shot?
2. Amy is about to begin her morning commute. Over her last 60 commutes, she arrived at work 12 times in under half an hour. What is the probability that she arrives at work in 3…

Classify each of the following probabilities as theoretical, empirical, or subjective.

1. An eccentric billionaire is testing a brand new rocket system. He says there is a 15% chance of failure.
2. With 4 seconds to go in a close basketball playoff game, the home team need 3 points to tie up the game and send it to overtime. A TV commentator says that team captain should take the final 3-point shot, be…