Question

A certain tennis player makes a successful first serve \(70 \%\) of the time. Assume that each serve is independent of the others. If she serves 6 times, what's the probability she gets a) all 6 serves in? b) exactly 4 serves in? c) at least 4 serves in? d) no more than 4 serves in?

Short answer

a) 0.117649. b) 0.324135. c) 0.74431. d) 0.579825.

Step-by-step solution

Understanding the Problem

We need to find probabilities related to the tennis player's serves. This is a binomial probability problem where each serve is an independent event with two possible outcomes: a successful serve or an unsuccessful serve. Given the probability of a successful serve, we need to determine various probabilities involving 6 attempted serves.

Identifying the Parameters

For a binomial distribution, we identify the number of trials, \( n = 6 \), and the probability of success, \( p = 0.7 \). We will use these parameters to calculate probabilities for different numbers of successful serves.

Calculating Probability for All 6 Serves In

To find the probability that she gets all 6 serves in, we calculate \( P(X = 6) \). The formula for the binomial probability is:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]For \( k = 6 \):\[ P(X = 6) = \binom{6}{6} (0.7)^6 (0.3)^0 = (0.7)^6 \approx 0.117649 \]

Calculating Probability for Exactly 4 Serves In

To find the probability of exactly 4 successful serves, use the formula with \( k = 4 \):\[ P(X = 4) = \binom{6}{4} (0.7)^4 (0.3)^2 \]Calculating gives:\[ \binom{6}{4} = 15 \]\[ P(X = 4) = 15 \times (0.7)^4 \times (0.3)^2 \approx 0.324135 \]

Calculating Probability for At Least 4 Serves In

To determine the probability of getting at least 4 successful serves, we find \( P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6) \).Calculate for \( k = 5 \):\[ P(X = 5) = \binom{6}{5} (0.7)^5 (0.3)^1 \approx 0.302526 \]Thus, \( P(X \geq 4) = 0.117649 + 0.302526 + 0.324135 = 0.74431 \).

Calculating Probability for No More Than 4 Serves In

The probability for no more than 4 successful serves is \( P(X \leq 4) \), which is equivalent to \( 1 - P(X \geq 5) \).Using our calculations:\( P(X \geq 5) = P(X = 5) + P(X = 6) = 0.302526 + 0.117649 = 0.420175 \)Thus, \( P(X \leq 4) = 1 - 0.420175 = 0.579825 \).

Key concepts

Probability Calculation

The essence of probability calculation in the context of a binomial distribution lies in determining the likelihood of a particular outcome. A binomial experiment typically involves a fixed number of trials, with each trial resulting in one of two outcomes: success or failure. The probability of success is consistent across all trials.

To calculate the probability for a specific number of successes, such as getting all 6 serves in or exactly 4 serves in, we use the binomial probability formula:

• \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Here, \( n \) refers to the total number of trials, \( p \) is the probability of a success in each trial, and \( k \) is the number of successful trials we are interested in.
Using this formula, we find insights into how likely an event is based on available data. For instance, the player's probability of getting all serves in, having exactly 4 successful serves, or other outcomes can be quantified precisely. This process builds a clear understanding of the likelihood of various outcomes in independent event scenarios.

Independent Events

Independent events in probability and statistics mean that the outcome of one event does not affect the outcome of another. This is a crucial concept in understanding binomial distributions. When events are independent, each trial is like starting fresh without any memory of previous outcomes.

For the tennis player's serves, this implies that whether a first serve is successful or not does not impact the success of the next serve. This independence allows us to model each serve using the same fixed probability of success (70% in this case).

• Being independent ensures consistency.
• Helps use real-world success rates efficiently in probabilistic models.

This concept underpins much of classical probability theory, ensuring that each serve in the exercise retains the same chance of success or failure regardless of past outcomes. Understanding this principle allows you to apply the binomial distribution effectively to real-life contexts like statistical modeling, gaming theory, or risk management.

Success-Failure Trials

In binomial distribution problems, trials can only result in one of two outcomes: success or failure. For the tennis player's serves, these outcomes are termed either a successful serve (success) or not (failure). Each serve she makes represents one trial in a series of such trials.

This concept is pivotal as it forms the basis of the binomial distribution. For any given problem:

• Define success probability \( p \).
• Determine failure probability as \( 1-p \).
• Analyze over a series of fixed, independent trials.

In our example, every time the player serves, there is a 70% chance of success and a 30% chance of failure. By focusing on success-failure trials, you can apply the straightforward formula for the binomial distribution to calculate any probability scenario, such as obtaining at least 4 successes from a series of 6 trials. Recognizing trials as binary helps simplify complex probability problems into manageable calculations.