Let’s Make A Deal:
Student Worksheet:
name: _________________
In Show Math, you saw an episode of the television show
Let’s Make a Deal! Are you convinced that switching doors
is the better strategy? Let’s see.
To simulate the game show, your team will have three playing cards: two black and one red. Red card = win, one of
the black cards = lose.
How to Play
1. One member of your team is the game show host. The host places all three cards face down on the table and
must remember exactly where each card is.
2. The other team member - the player - chooses a card by pointing at it.
3. The host flips over one black card from the two remaining cards—if both remaining cards are black it doesn’t
matter which one is flipped.
4. The player then decides whether to keep the original card or switch to the other unseen card.
5. Finally, the cards are revealed and it is determined whether the player has won or lost and the result recorded
in a table.
1. Record your results on these tables. Write W when you win and L when you lose. Make sure you play each
strategy 15 times to fill in the table.
Strategy 1: Always keep the same card
#1
#2
#3
#4
#5
#6
#7
#8
#9
#10
#11
#12
#13
#14
#15
Strategy 2: Always switch cards
#1
#2
#3
#4
#5
#6
#7
#9
#10
#11
#12
#13
#14
#15
#8
2. How many times did you win when you kept the same card? _______
How many times when you switched cards? ______
3. What was the experimental probability of winning for each strategy?
Group results
Strategy 1: Keep the same card on
every turn
Experimental probability
Group results
Strategy 2: Switch cards on every
turn
Experimental probability
4. Calculate the results for the class as a whole by filling in the table below.
Class results
Strategy 1: Keep the
same card on every
turn
Strategy 2: Switch
cards on on every
turn
Total #
of successes
Total #
of trials
Experimental
probability
Experimental probability
An experiment is an action involving chance that leads to a result (eg.
flipping a coin). Each repetition of
the experiment is called a trial. The
experimental probability is the ratio
of the number of successes (eg. getting Heads, H, when flipping a coin)
to the number of trials.
Experimental Probability=
Number of Wins
Number of Attempts
5. Now let’s determine the theoretical probability of winning for each
strategy. We do this by listing all the possible outcomes of the game
using the tree diagram below. First do this for Strategy 1. The first
branch specifies the prize door. The next branch is the door that you
picked. For example if you follow the first branch of the tree diagram,
you see that the prize door is 1, and then you follow it to the next
branch and you see that the door picked is 1, thus you win and would
fill in a W. Fill in the next eight spots with W or L as appropriate.
Theoretical probability
An outcome is the result of a
single trial of an experiment.
The possible outcomes of an experiment is the list of all the
different outcomes that could
have occurred during the experiment (eg. the four possible
outcomes of flipping a coin two
times are: HH, HT, TH, and TT).
The theoretical probability is the
ratio of the number of successes
to the number of possible
outcomes.
Theoretical Probability =
Number of wins
Number of possible outcomes
6. From this calculate the theoretical probability of winning using the stay
strategy.
7. Now let us examine the different possibilities when using the switching strategy. We will again use a tree
diagram. The first branch specifies the prize door and the second branch represents the door that is initially
chosen. You must fill in what door Monty Hall will reveal. Assume that if Monty has two goat doors to choose
from, that he will reveal the lower numbered door. It doesn’t affect the probabilities. Then fill in the number
of the door you switch to and then whether or not you win or lose. For example, when the prize is behind
Door #1, and you pick Door #1, Monty will reveal Door #2 and you will switch to Door #3 and lose.
8. Calculate the theoretical probability of winning for the switching strategy.
9. Compare the theoretical probability and your group’s experimental probability. Also compare the theoretical
probability and the class experimental probability. Which was closer to the theoretical probability, your value
or the class value?
Student Take Home Worksheet
1. As you have seen, we can sometimes judge too quickly. Suppose there are two bags and each bag contains
two marbles. Bag A contains one white marble and one black marble. Bag B contains two white marbles.
Now suppose you choose one of the bags at random without know what is in it. Then, you draw a white
marble. What is the probability that the remaining marble in the bag is white? Explain your results.
[Hint: Start by drawing a tree diagram to show all the possible outcomes of choosing bags and
then marbles.]
2. Take a look at another counterintuitive probability problem. You are told that Mr. Jones has two children
and the older child is girl.
a. What is the probability that both children are girls?
b. Now what if you are told that Mr. Jones has two children and you know that one is a girl (with no age
specified), what is the probability that other child is a girl?
[Hint: Draw a tree diagram FIRST]