Lottery. In Exercises 15–20, refer to the accompanying table, ...

Question

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).

Range Rule of Thumb for Significant Events Use the range rule of thumb to determine whether 4 matches is a significantly high number of matches.

Table showing probabilities for matching digits in a lottery: 0 (0.656), 1 (0.292), 2 (0.049), 3 (0.004), 4 (0+).

Range Rule of Thumb for Significant Events Use the range rule of thumb to determine whether 4 matches is a significantly high number of matches.

Accepted Answer

In a security contest, participants guess a four digit code. Each digit ranges from 0 to 9, and digits can repeat. The winning code is generated randomly. The random variable X represents the number of digits guessed correctly in the exact position. The following table shows the probabilities of correctly matching 0123, or 4 digits in correct positions. And we're given a probability table. Use the range rule of thumb to determine whether getting all four digits correct is a significantly high number of matches. And we have 3 possible answers, being yes, no, or the rule can't be applied. Now, to solve this, let's first find the mean. The mean is given by. The sum of our number of matches multiplied by their probabilities. In this case, this will be 0, multiplied by 0.6561. Plus 1, multiplied by 0.2916. Plus 2 multiplied by 0.0486. Plus 3, multiplied by 0.0036. Plus 4 multiplied by 0.0001. We can simplify this to get the value of 0.4. Now, let's also find the standard deviation. We can find standard deviation by first finding the variant sigma squared. Equals Yeah X Minus musesquirt. We get 0. 5200 minus our mean. 0.4 squared. This gives us the value 0.36. Which means our standard deviation is a square root of 0.36 or just 0.6. Now, let's apply the range rule of thumb. This tells us the outcome is significant. If X is greater than equals to, I mean. Plus 2 standard deviations. We'll then have X is greater or equals to mean 0.4 + 2 multiplied. By 0.6. This means X should be greater than or equals to 1.6 to be significant. Now, in our case, we have 4 matches, which is X equals 4. 4 is indeed greater than 1.6. Because this is greater than our value. We will say yes. By getting 4 matches. is significantly a high number based on their range rule of thumb. This then corresponds to answer A. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Key Concepts

Probability Distribution

Random Variable

Range Rule of Thumb

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