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Names _____________________________
_____________________________
_____________________________
Multi-stage Probabilities
Part One. Suppose you were to do the following: Flip a coin. Observe heads or tails. Flip it
again and observe heads or tails.
1. What is the probability that the first toss results in a head? ____ What are you assuming?
2. What is the probability that the second toss results in heads? ____ Does it matter whether the
first toss was heads or tails? _____
3. Heads followed by tails (HT) is one possible sequence. List all the possible sequences.
4. These sequences can be written in the form of a “tree diagram”. For the first toss, list the
possibilities and their chances. Then for the second, list the possible results and their chances.
Fill in the blanks in the following tree diagram.
1st Toss
H
prob. =
T
prob. =
2nd Toss
H
prob. =
T
prob. =
Result
HH
H
prob. =
T
prob. =
TH
HT
TT
5. What do you think are the probabilities for each of these four results? Are they the same, or
are some of the results more likely than others? If different, how?
Result:
HH
HT
TH
TT
Prob. Guess:
6. In any case, what must the probabilities of the four results add up to? ______
7. Take a coin and carry out this “experiment” 20 times. That is, do 20 repetitions of the two-flip
experiment. Keep a tally of the number of times each sequence occurred.
Result:
HH
HT
TH
TT
Tally:
Fraction:
8. What fraction of the time did each result occur? Write the fraction above in decimal notation.
Did your fractions match the probabilities that you projected in Part 5? Should they?
9. Your instructor will compile all the tallies for the class. Do you expect these to be closer to
the values you anticipated in #5 than what you observed for your 20 repetitions of the
experiment? ____ Are they?
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Part Two. Here is another experiment: Two red cards and two black cards are shuffled. A card
is dealt to you. Then another card is dealt to you so that you now have two cards.
10. What is the probability that the first card is red? ______
11. Suppose your first card is red and you are about to be dealt the second card. Now what is the
probability that the second card will be red? _____ In this case, what is the probability that the
second card will be black? _____
12. Suppose your first card is black, and you are about to be dealt the second card. In this
scenario, what is the probability that the second card is red? ______ black? ______
_______ Stop here and wait until we go over our answers to #10-12. ___
13. Write these probabilities in the following tree diagram.
1st Card
R
prob. =
B
prob. =
2nd Card
R
prob. =
B
prob. =
Result
RR
R
prob. =
B
prob. =
BR
RB
BB
14. Speculate as to the probabilities of the four possible results.
Result:
RR
RB
BR
Prob. Guess:
BB
15. What must the probabilities of the four results add up to? ______
18. Take two red cards and two black cards and repeat this “experiment” 20 times, shuffling the
four cards well in between each round of dealing. One partner can do the shuffling and dealing,
and the other can tally the results below.
Result:
RR
RB
BR
BB
Tally:
Fraction:
19. Write the fraction of the time that each sequence occurred above, using decimal notation.
Did these match well with you projections (#14)?
20. Your instructor will compile all the tallies for the class. Do you expect these to be closer to
the values you anticipated in #14 than what you observed for your 20 repetitions of the
experiment? ____ Are they?
_______ Stop here and wait until we go over our answers through #20. ___
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21. Hopefully you have noticed a pattern by now. How can the final result probabilities be found
from the probabilities at each stage?
_______ Stop here and wait until we go over our answers to #21. ___
Part Three. Make a tree diagram and figure out the resulting probabilities for each of the
following problems. In each case, use the property that you noticed and reported on in #21.
22. A married couple is planning to have children. Suppose they will end up with three kids.
Assume that each child is just as likely to be a girl as a boy. Figure out the possible sequences,
such as BGG, as well as the probabilities for each sequence. Make a tree diagram.
23. Three red cards and two black cards are shuffled together. Two are dealt. Figure the
probabilities of the four possible sequences.
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Part Four. Use the property that you reported on in #21 to figure out the following probabilities.
(Don’t make the tree diagram – they are humongous in these cases!)
24. A regular deck of 52 cards is shuffled and two cards are dealt to you. What is the probability
that both are aces?
25. A balanced six-sided die is rolled twice. What is the probability that it comes up “6” both
times?
26. Two balanced dice are rolled together. What is the probability that they are both 6’s?
27. A student is taking a quiz. The first question is True/False and the second is Multiple Choice
with one answer to be selected from A, B, C or D. Unfortunately, the student hasn’t studied and
is totally clueless. What is the probability that this student will get both answers correct?
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