Card Sharp

In this activity, the students calculate probabilities and mark them on a number line. This helps them to visually compare the probabilities of various outcomes. The main concept (and one that is likely to be new to many students) is that the word “or” always increases the probability of success (because the number of acceptable outcomes is increased), whereas “and” always reduces the probability of success (because there are now more conditions to satisfy). An explanation of the principle is given in the Answers for question 4.

Although the students are told to work with a classmate, they could equally well work by themselves and then discuss their answers. You should have a pack of cards available for students to refer to.

When the students are counting the possible outcomes for an event, they will need to take care not to double-count any. For example, when they are considering the probability of getting a red card or an ace, they will find there are 26 red cards and 4 aces (a total of 30). However, because there are 2 red aces (cards that are red and aces), these cards must only be counted once.

When doing question 1, the students need to draw a number line that goes from 0 to 1. They can divide this line into either a multiple of 10 parts or a multiple of 13 parts. If they use a line with 10 or 20 divisions, they will need to convert the fractions to decimals before they can locate an event on the line. If they use 13, 26 or 52 divisions, they will find the placement easier, but they may have trouble understanding that all these awkward-looking fractions lie between 0 and 1.

In the activity, the card is returned and the pack is re-shuffled before another card is taken. Interested students may like to explore the situation in which the card is not returned to the pack before another is taken. They could investigate this by using a small number of cards (for example, 2 aces and 3 kings) and looking at what happens when they draw 2 cards without replacing them. They could then consider the probability of getting 2 aces in a row, starting with a full pack. If the first card is returned before the second is taken, the probability is 4/52 x 4/52 = 16/2704 = 0.0059. If the first card is not returned, the probability is 4/52 x 3/51 = 12/2652 = 0.0045. The probability of getting 2 aces in a row is therefore greater if the first card is replaced before the second is taken.

Answers to Activity

1. a. 7: 0.08 (4/52); red spade: 0 (0/52); 4 or heart: 0.31 (16/52); red or ace: 0.54 (28/52); even number: 0.38 (20/52); king or queen: 0.15 (8/52); black 1, 2, or 3: 0.12 (6/52); a red or a black: 1.0 (52/52).

b. Answers will vary.
2. a. 0.08 (4/52), 0.08 (4/52), 0.15 (8/52)
b. 0.08 (4/52), 0.25 (13/52), 0.31 (16/52)
c. 0.08 (4/52), 0.25 (13/52), 0.5 (26/52), 0.77 (40/52)
3. “Or” means that the number of successful outcomes is increased (because there are more ways of winning), so the use of “or” always pushes the probability closer to 1.
4. The use of “and” always decreases the probability of success, that is, it pushes the probability closer to 0. This is  because success is being made dependent on meeting further conditions. Someone who is tall and blond and blue-eyed and left-handed is much harder to find than someone who is just tall or just left-handed.