4. Suppose there is a card game where you are dealt a hand of three cards. You have already learned that the total number of three-card hands that can be dealt from a deck of 52 cards is: 

In this problem you will calculate the probability of getting a hand that has exactly two aces in it (A A X). You will do this by finding out the number of possible hands that have exactly two aces, and then dividing by the total possible number of three-card hands that is stated above.

Part A: Use the multiplication principle to tell the total number of three-card hands (permutations) that can be made with two aces. (2 points)

Part B: In the answer from Part I, each two-ace hand got counted twice. For example, A A X got counted as a separate hand from A A X. Since order should not matter in a card hand, these are really the same hand. What is the actual number of two-ace hands (combinations) you can get from a deck of 52 cards?(2 points)

Part C: Find the probability of drawing a three-card hand that includes two aces from a deck of 52 cards. Write your answer as a fraction. (2 points)

5. A 2011 study by The National Safety Council estimated that there are nearly 5.7 million traffic accidents year. At least 28% of them involved distracted drivers using cell phones or texting. The data showed that 11% of drivers at any time are using cell phones . Car insurance companies base their policy rates on accident data that shows drivers have collisions approximately once every 19 years. That’s a 5.26% chance per year. Given what you know about probability, determine if cell phone use while driving and traffic accidents are related.

Step A: Let DC = event that a randomly selected driver is using a cell phone. What is P(DC)? (1 point)

Step B: Let TA = event that a randomly selected driver has a traffic accident. What is P(TA)? Hint: What is the probability on any given day? (1 point)

Step C: How can you determine if cell phone use while driving and traffic accidents are related? (1 point)

Step D: Given that the driver has an accident, what is the probability that the driver was distracted by a cell phone? Write this event with the correct conditional notation. (1 point)

Step E: What is the probability that a randomly selected driver will be distracted by using a cell phone and have an accident? (2 points)

Step F: For a randomly selected driver, are the events “driving while using a cell phone” and “having a traffic accident” independent events? Explain your answer. (2 points)