Example 2.29 from the textbook. A 12-unit apartment building has 9 smokes alarms which pass inspection and 3 which do not. If 3 smoke alarms are selected at random and tested, what is the probability that all pass inspection. (a) How many ways is there to select 3 smokes alarm? (b) How many way is there to only select alarms which pass the test? (c) What is the probability to select alarms which pass the test?
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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Example 2.29 from the textbook. A 12-unit apartment building has 9 smokes alarms which pass inspection and 3 which do not. If 3 smoke alarms are selected at random and tested, what is the
probability that all pass inspection.(a) How many ways is there to select 3 smokes alarm?
(b) How many way is there to only select alarms which pass the test?
(c) What is the probability to select alarms which pass the test?
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Suppose you have 7 coins in your pocket: 4 dimes and 3 quarters. 2 coins are selected simultaneously and at random.
(a) How many ways are there to select 2 coins?
(b) How many ways are there to pick 2 quarters?
(c) What is the probability that you pick 2 quarters?
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You draw 4 cards in a deck of 52 cards. We want to find the probability that your hand contains more than one suit. (Recall that there are 4 suits, each containing 13 cards)
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(a) How many different hands can you get?
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(b) How many different hands containing more than one suit can you get? (Hint: Think about the complement)
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(c) What is the probability that your hand contain more than one suit?
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Let’s rework example 2.32 from you textbook, but with different numbers. A 8 person committee consist of 5 women, including Margaret, and 3 men, including Anthony. If a subcommittee of 3 people is to be chosen at random to evaluate a new project. The subcommittee needs to include at least 1 women, but can’t consist entirely of women.
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(a) How many ways are there to select a subcommittee which consist of 1 man and 2 women?
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(b) How many ways are there to select a subcommittee which consist of 2 man and 1 women?
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(c) How many subcommittees satisfying the conditions can be selected?
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(d) How many subcommittees which include Anthony can be selected?
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(e) What is the probability that Anthony is included in the subcommittee
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(f) How many different subcommittees include Margaret?
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(g) How many different subcommittees include at least one of Anthony or Margaret? (Hint: Make sure that you do not count a committee more than once)
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(h) What is the probability that at least one of Anthony or Margaret are in the subcommittee?
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Suppose that you flip a fair coin 6 times in a row and record each results. We want to find the probability that you get both heads and tails.
(a) How many different outcomes can you get?
(b) How many different outcomes contains both heads and tails?(c) What is the probability that you got both heads and tails?
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