Players in sports are said to have "hot streaks" and "cold streaks." For example, a batter in baseball might be considered to be in a slump, or cold streak, if that player has made 10 outs in 10 consecutive at-bats. Suppose that a hitter successfully reaches base 42% of the time he comes to the plate. Complete parts (a) through (c) below. (a) Find the probability that the hitter makes 10 outs in 10 consecutive at-bats, assuming at-bats are independent events. Hint: The hitter makes an out 58% of the time. P(hitter makes 10 consecutive outs) = (Round to five decimal places as needed.) (b) Are cold streaks unusual? The probability of a cold streak is V 0.05, so cold streaks V unusual. (c) Interpret the probability from part (a). In repeated sets of 10 consecutive at-bats, the hitter is expected to make an out in all 10 at-bats about times out of 1000. (Type a whole number.)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**Title: Understanding "Hot Streaks" and "Cold Streaks" in Sports Statistics**
Players in sports are often described as having "hot streaks" and "cold streaks." For instance, a batter in baseball is considered to be in a slump, or cold streak, if the player has made 10 outs in 10 consecutive at-bats. Assume that a hitter successfully reaches base 42% of the time he comes to the plate. Solve parts (a) through (c) below:
**(a) Probability Calculation**
Find the probability that the hitter makes 10 outs in 10 consecutive at-bats, assuming at-bats are independent events.
*Hint:* The hitter makes an out 58% of the time.
\[
\text{P(hitter makes 10 consecutive outs) = } \boxed{\phantom{answer}}
\]
*Round to five decimal places as needed.*
**(b) Unusual Nature of Cold Streaks**
Are cold streaks unusual?
The probability of a cold streak is \(\boxed{\phantom{answer}}\), with a threshold of 0.05, indicating whether cold streaks are \(\boxed{\text{unusual/not unusual}}\).
**(c) Interpretation of Probability**
Interpret the probability from part (a).
In repeated sets of 10 consecutive at-bats, the hitter is expected to make an out in all 10 at-bats about \(\boxed{\phantom{answer}}\) times out of 1000.
*Type a whole number.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26e9d1e4-e751-46db-8f1e-f1957aec9903%2F4627f85e-0f3e-4331-ba4c-a258402b56b0%2Flaugsbk_processed.jpeg&w=3840&q=75)
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