Suppose you and a friend are playing a game that involves flipping a fair coin 3 times. Let [tex]$X=$[/tex] the number of time that the coin shows heads. The probability distribution of [tex]$X$[/tex] is shown in the table.
| Number of Heads | 0 | 1 | 2 | 3 |
| :---------------- | :---- | :---- | :---- | :---- |
| Probability | 0.125 | 0.375 | 0.375 | 0.125 |
In determining if this is a binomial setting, how has the binary condition been met?
A. "Success" = the coin shows tails. "Failure" = the coin shows heads.
B. "Success" = the coin shows heads. "Failure" = the coin shows tails.
C. "Success" = the coin shows 1,2, or 3 heads. "Failure" = the coin shows 0 heads.
D. "Success" = the coin shows all 3 heads. "Failure" = the coin shows 0, 1, or 2 heads.
Answer (2)
The binary condition in a binomial setting is met when each trial has only two outcomes: success or failure. The options that correctly define this for a single coin flip are:
Success = the coin shows tails, Failure = the coin shows heads.
Success = the coin shows heads, Failure = the coin shows tails.
Explanation
Understanding the Binary Condition In a binomial setting, the binary condition requires each trial to have only two possible outcomes: success or failure. We need to determine which of the given options correctly defines 'success' and 'failure' for a single coin flip.
Analyzing Option 1 Option 1: "Success" = the coin shows tails. "Failure" = the coin shows heads. This option correctly defines two outcomes for a single coin flip.
Analyzing Option 2 Option 2: "Success" = the coin shows heads. "Failure" = the coin shows tails. This option also correctly defines two outcomes for a single coin flip.
Analyzing Option 3 Option 3: "Success" = the coin shows 1, 2, or 3 heads. "Failure" = the coin shows 0 heads. This option defines success and failure based on the number of heads in 3 flips, not a single trial. Therefore, it does not meet the binary condition for a single trial.
Analyzing Option 4 Option 4: "Success" = the coin shows all 3 heads. "Failure" = the coin shows 0, 1, or 2 heads. Similar to option 3, this option defines success and failure based on the number of heads in 3 flips, not a single trial. Thus, it does not meet the binary condition for a single trial.
Conclusion Therefore, options 1 and 2 correctly define the binary condition for a single coin flip in a binomial setting.
Examples
Consider a quality control process where you inspect items for defects. Each item is either defective (success) or not defective (failure). The binomial setting helps calculate the probability of finding a certain number of defective items in a batch.
The binary condition for a binomial setting in this context is met by designating one outcome as 'success' and the other as 'failure'. The correct answer is option B: 'Success' = the coin shows heads, 'Failure' = the coin shows tails. This clearly meets the requirement of having two possible outcomes for each trial.
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