Question 1190869: For each of the following situations, state the probability rule or rules that you would use and apply it or them. Write a sentence explaining how the situation illustrates the use of the probability rules.
a)A coin is tossed three times. The probability of zero heads is 1/8, and the probability of zero tails is 1/8. What is the probability that all three tosses result in the same outcome?
b)Refer to part (a). What is the probability that there is at least one head and at least one tail?
c)The probability of event A is 0.5, and the probability of event B is 0.9. Events A and B are disjoint. Can this happen?
d)Event A is very rare. Its probability is −0.04. Can this happen?
e)The probability of event A is 0.23. What is the probability that event A does not occur?
f)You toss a coin two times. It is not a fair coin, and you do not know the probability of a head. What is the probability that either zero, one, or two heads appear?
Answer by math_tutor2020(3816)   (Show Source):
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Part (a)
Add the fractions mentioned
1/8 + 1/8 = 2/8 = 1/4
The probability of getting all the same outcome (either all heads or all tails) is 1/4
Answer: 1/4
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Part (b)
H = heads
T = tails
The previous probability we calculated is complementary to this current probability we're after.
Either we get all the same result (HHH or TTT) or we get some mix of H's and T's.
So we subtract the previous result from 1.
1 - (1/4) = 3/4
Answer: 3/4
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Part (c)
If A and B are disjoint and the only possible events to occur, then,
P(A or B) = P(A) + P(B)
However, we have
P(A or B) = P(A) + P(B)
P(A or B) = 0.5 + 0.9
P(A or B) = 1.4
which is larger than 1.
We cannot have probability values larger than 1
The number 1 itself means "100% certainty".
We cannot exceed this (how can we be more certain than the most certain possible?)
All probability values must be between 0 and 1 inclusive
 \le 1)
Answer: No it can't happen
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Part (d)
Same idea as part (c).
The probability value cannot be negative as it's outside of the interval
If you want to describe some (very) rare event, then use small positive values close to 0.
A probability of zero itself means "impossible" or "certain not to occur".
Answer: No it's not possible.
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Part (e)
Subtract the probability from 1
1 - 0.23 = 0.77
If an event has a 23% chance of occurring, then it has a 77% chance of not occurring.
Answer: 0.77
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Part (f)
It doesn't matter what the probability of getting heads is.
The description of "either zero, one, or two heads appear" describes every possible outcome. One of those events will happen.
This is the entire sample space
HH
HT
TH
TT
which shows symbolically what was described in the previous paragraph. Clearly there's no way to get anything other than what's described, so we know 100% that we'll get one of those things happening.
We indicate 100% probability with the number 1. This is to contrast 0 which means "impossible".
Answer: 1
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