Our decisions depend on how the options are presented to us. Here's an experiment that illustrates this phenomenon. Tell 20 subjects that they have been given $50 but can't keep it all. Then present them with a long series of choices between bets they can make with the $50. Scattered among these choices in random order are 64 choices between a fixed amount and an all-or-nothing gamble. The odds for the gamble are always the same, but 32 of the fixed options read "Keep $20" and the other 32 read "Lose $30." These two options are exactly the same except for their wording, but people are more likely to gamble if the fixed option says they lose money. Here are the percent differences ("Lose $30" minus "Keep $20") in the numbers of trials on which the 20 subjects chose to gamble: 37.5 30.8 6.2 17.6 14.3 8.3 16.7 20.0 10.5 21.7 30.8 27.3 22.7 38.5 8.3 10.5 8.3 10.5 25.0 7.7 If we make a stemplot, we would find the stemplot shows a slight skew to the right, but not so strong that it would invalidate the t procedures. All 20 subjects gambled more often when faced with a sure loss than when faced with a sure win. Give a 99% confidence interval for the mean percent increase in gambling when faced with a sure loss. __to__
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.
Our decisions depend on how the options are presented to us. Here's an experiment that illustrates this phenomenon. Tell 20 subjects that they have been given $50 but can't keep it all. Then present them with a long series of choices between bets they can make with the $50. Scattered among these choices in random order are 64 choices between a fixed amount and an all-or-nothing gamble. The odds for the gamble are always the same, but 32 of the fixed options read "Keep $20" and the other 32 read "Lose $30." These two options are exactly the same except for their wording, but people are more likely to gamble if the fixed option says they lose money. Here are the percent differences ("Lose $30" minus "Keep $20") in the numbers of trials on which the 20 subjects chose to gamble:
| 37.5 | 30.8 | 6.2 | 17.6 | 14.3 | 8.3 | 16.7 | 20.0 | 10.5 | 21.7 |
| 30.8 | 27.3 | 22.7 | 38.5 | 8.3 | 10.5 | 8.3 | 10.5 | 25.0 | 7.7 |
If we make a stemplot, we would find the stemplot shows a slight skew to the right, but not so strong that it would invalidate the t procedures. All 20 subjects gambled more often when faced with a sure loss than when faced with a sure win. Give a 99% confidence interval for the mean percent increase in gambling when faced with a sure loss.
__to__
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