Stats - Chapter 5

If the probability that it will rain tomorrow is​ 0.30, the probability that it will not rain tomorrow is​ what?

0.70 The probability that an event does not occur is 1 minus the probability that the event will occur. The probability that it will not rain tomorrow is then 1−0.30=0.70.

In​ statistics, what is true of​ randomness?

In​ statistics, randomness is hard to achieve without help from a computer or some other randomizing​ device, such as a random number table.

In a simulation of coin​ tosses, a streak of 15 heads has appeared. The Law of Large Numbers says which of the following must be​ true?

It is equally likely that the 16th toss will be a head or a tail. The Law of Large Numbers is patient. It says that the empirical probability will equal the true probability after infinitely many trials. A streak of 10 or 20 or even 100​ heads, though extremely​ rare, does not contradict the LLN.

Experiments used to produce empirical probabilities are called​ what?

Simulations Experiments used to produce empirical probabilities are called​ simulations, because the investigators hope that these experiments simulate the situation they are examining.

Which of the following is the probability that something in the sample space will​ occur?

1 The sample space is the set of all possible and equally likely outcomes of the experiment. Because it contains all possible​ outcomes, the probability that something in the sample space will occur is 1.

Which of the following might be a reason that an empirical probability found through a simulation does not match the theoretical​ probability?

1. The empirical value is just varying. Doing more trials will get closer to the theoretical probability. 2. The theoretical value is incorrect. People are not​ infallible; the theroetical value may be incorrect due to a miscalculation. In​ addition, empirical values vary. Doing more trials will get closer to the theoretical probability.

In most​ cases, it is recommended that at least how many trials be done when using a simulation to estimate a​ probability?

100 In most​ cases, it is recommended that least 100 trials be done when using a simulation to estimate a probability.

Probabilities that are based on​ short-run relative frequencies are called​ what?

Empirical probabilities Probabilities that are based on​ short-run relative frequencies are called empirical probabilities. If a coin is tossed 10 times and 6 results are​ heads, the empirical probability of getting heads is 610=​0.6, or​ 60%.

A friend flips a coin 10 times and says that the probability of getting a head is 40​% because he got four heads. Is the friend referring to an empirical probability or a theoretical​ probability? Explain.

Empirical probabilities are​ short-run relative frequencies based on an experiment.

What are probabilities?

Long-run relative frequencies used to describe processes where the outcome depends on chance, such as flipping a coin or rolling a die.

When two events have no outcomes in​ common, they are called​ what?

Mutually Exclusive When two events have no outcomes in​ common, they are called mutually exclusive. Rolling a 5 on a​ six-sided die and rolling a 3 on a​ six-sided die are mutually exclusive events.

What does random mean in statistics?

No predictable pattern occurs and no digit is more likely to occur than any other.

Statistics and probability use the​ "inclusive OR". This means that referring to outcomes A OR B is referring to​ what?

Outcomes that are only in​ A, only in​ B, or in both

Probabilities are always numbers between and including what​ numbers?

Probabilities are always numbers between and including 0 and 1. If the probability of an event happening is​ 0, then that event never happens. If the probability of an event happening is​ 1, then that event always happens.

What are empirical probabilities?

Probabilities based on observation of the actual process.

What are theoretical probabilities?

Probabilities based on specific assumptions (usually based on a theory) about the chance process.

If an experiment with a random outcome is repeated a large number of​ times, the empirical probability of an event is likely to be close to the true probability. This mathematical theorem is called​ what?

The Law of Large Numbers The Law of Large Numbers states that if an experiment with a random outcome is repeated a large number of​ time, the empirical probability of an event is likely to be close to the true probability. The larger the number of​ repetitions, the closer together these probabilities are likely to be.

Given the event​ "a die lands with a 6 on​ top", which of the following is the complement of this​ event?

The complement is all the ways the die can land with a number that is not a 6 on top. This happens when the die lands with a​ 1, 2,​ 3, 4, or 5 on top.

The sample space of a random experiment is​ what?

The sample space of a random experiment is the set of all possible and equally likely outcomes of the experiment. It is often represented with the letter S.

A Monopoly player claims that the probability of getting a 4 when rolling a​ six-sided die is 16 because the die is equally likely to land on any of the six sides. Is this an example of a theoretical probability or an empirical​ probability? Explain.

Theoretical probability is the relative frequency at which an event happens after infinitely many repetitions. This value never changes.

Because they are generated by a seed value that starts the random​ sequence, computer-generated random numbers are sometimes called​ what?

​pseudo-random Computer-generated random numbers are sometimes called​ pseudo-random numbers. If a researcher inputs the same seed​ number, that researcher will always see the same sequence of​ pseudo-random numbers.