In: Math
Please answer form 6-14
I. Consider the random experiment of rolling a pair of dice. Note: Write ALL probabilities as reduced fractions or whole numbers (no decimals).
1) One possible outcome of this experiment is 5-2 (the first die comes up 5 and the second die comes up 2). Write out the rest of the sample space for this experiment below by completing the pattern:
1-1 |
2-1 |
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1-2 |
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1-3 |
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1-4 |
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1-5 |
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1-6 |
2) How many outcomes does the sample space contain? _____________
3) Draw a circle (or shape) around each of the following events (like you would to circle a word in a word search puzzle). Label each event in the sample space with the corresponding letter. Event A has been done for you.
A: Roll a sum of 3.
B: Roll a sum of 7.
C: Roll a sum of at least 10.
D: Roll doubles.
E: Roll snake eyes (two 1’s). F: First die is a 4.
4) Find the following probabilities:
P(A) = _________ P(B) = _________ P(C) = _________
P(D) = _________ P(E) = _________ P(F) = _________
5) The conditional probability of B given A, denoted by P(B|A), is the probability that B will occur when A has already occurred. Use the sample space above (not a special rule) to find the following conditional probabilities:
P(D|C) = _________ P(E|D) = _________ P(D|E) = _________ P(A|B) = _________ P(C|F) = _________
6) Two events are mutually exclusive if they have no outcomes in common, so they cannot both occur at the same time.
Are C and E mutually exclusive? ___________
Find the probability of rolling a sum of at least 10 and snake eyes
on the same roll, using the
sample space (not a special rule).
P(C and E) = __________
Find the probability of rolling a sum of at least 10 or snake eyes, using the sample space. P(C or E) = __________
7) Special case of Addition Rule: If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B)
Use this rule to verify your last answer in #6:
P(C or E) = P(C) + P(E) = ________ + ________ = _________
8) Are C and F mutually exclusive? __________ Using sample space, P(C or F) = _________ 9) Find the probability of rolling a “4” on the first die and getting a sum of 10 or more, using the
sample space.
P (C and F) = ________
10) General case of Addition Rule: P(A or B) = P(A) + P(B) – P(A and B) Use this rule to verify your last answer in #8:
P(C or F) = P(C) + P(F) – P(C and F) = ________ + ________ − ________ = _________
11) Two events are independent if the occurrence of one does not influence the probability of the other occurring. In other words, A and B are independent if P(A|B) = P(A) or if P(B|A) = P(B).
Compare P(D|C) to P(D), using the sample space: P(D|C) =
________ . P(D) = ________ .
Are D and C independent? _________
When a gambler rolls at least 10, is she more or less likely to
roll doubles than usual? ___________ Compare P(C|F) to P(C), using
the sample space: P(C|F) = ________ . P(C) = ________ .
Are C and F independent? __________
12) Special case of Multiplication Rule: If A and B are
independent, then P(A and B) = P(A) · P(B).
Use this rule to verify your answer to #9:
P(C and F) = P(C) • P(F) = ________ · ________ = ________ .
13) Find the probability of rolling a sum of at least 10 and getting doubles, using the sample space. P(C and D) = ________ .
14) General case of Multiplication Rule: P(A and B) = P(A) · P(B|A). Use this rule to verify your answer to #13:
P(C and D) = P(C) • P(D|C) = ________ · ________ = ________ .