Probability of making 2 shots in 6 attempts | Probability and Statistics | Khan Academy
TLDRThe video script explores the probability of scoring exactly two free throws out of six attempts, given a 70% success rate. It explains the concept of complementary probabilities, where the chance of missing is 30%. The script then delves into combinatorics, calculating the number of ways to achieve two scores using 'six choose two', and multiplies this by the probability of each specific scenario. The final calculation, using the binomial coefficient and the given probabilities, results in approximately a 6% chance of scoring exactly two out of six free throws, highlighting the relatively low likelihood of this outcome with a high success rate.
Takeaways
- 🏀 The probability of scoring a free throw is given as 70% or 0.7 in decimal form.
- 🎯 The probability of missing a free throw is calculated as 100% - 70%, which equals 30% or 0.3.
- 📚 The sum of the probabilities for scoring and missing must total 100% or 1, reflecting the only two possible outcomes.
- 🔢 The scenario involves calculating the probability of scoring exactly two out of six free throw attempts.
- 🎲 The script explores different specific sequences of scoring exactly two out of six attempts and their respective probabilities.
- 🧩 Each sequence's probability is calculated by multiplying the probabilities of individual outcomes (0.7 for score and 0.3 for miss).
- 📈 The total number of ways to score exactly two out of six attempts is determined using combinatorics, specifically 'six choose two'.
- 🔑 The formula for combinations is applied to find the number of ways to choose two successful attempts out of six, which equals 15.
- 📉 The overall probability of scoring exactly two out of six attempts is the product of the individual sequence probability and the number of such sequences.
- 🧮 The final calculation yields a probability of approximately 0.05935, or about 5.93%, when rounded to the nearest percentage.
- 🤔 The script concludes by noting that with a high free throw percentage, it is actually quite unlikely to score exactly two out of six attempts.
Q & A
What is the probability of scoring a free throw according to the script?
-The probability of scoring a free throw is stated to be 70% or 0.7 when expressed as a decimal.
How is the probability of missing a free throw calculated in the script?
-The probability of missing a free throw is calculated by subtracting the probability of scoring from 100%, which results in 30% or 0.3.
What is the total number of free throw attempts considered in the script?
-The script considers a total of six free throw attempts.
What is the script trying to calculate?
-The script is trying to calculate the probability of exactly two scores in six free throw attempts.
How many different ways can two scores occur in six attempts according to the script?
-There are 15 different ways that two scores can occur in six attempts, as calculated by the binomial coefficient 'six choose two'.
What is the formula used to calculate the number of ways to get two scores in six attempts?
-The formula used is the binomial coefficient, which is calculated as 'six choose two' or 6! / (2! * (6-2)!), where '!' denotes factorial.
What is the significance of the term '0.7 squared' in the probability calculation?
-The term '0.7 squared' represents the probability of scoring on two consecutive attempts, given the probability of scoring on any single attempt is 0.7.
What does '0.3 to the fourth power' signify in the script?
-'0.3 to the fourth power' signifies the probability of missing four consecutive attempts, given the probability of missing any single attempt is 0.3.
How is the final probability of exactly two scores in six attempts calculated?
-The final probability is calculated by multiplying the number of ways to get two scores (15) by the probability of any particular way occurring (0.7 squared times 0.3 to the fourth power).
What is the approximate probability of getting exactly two scores in six attempts according to the script?
-The approximate probability of getting exactly two scores in six attempts is about 6%.
Why might the calculated probability of exactly two scores be considered low?
-The calculated probability is considered low because the given free throw percentage (70%) is relatively high, making it less likely that only two scores would be made in six attempts.
Outlines
🏀 Calculating the Probability of Scoring in Basketball Free Throws
The paragraph explains the concept of calculating probabilities in a basketball free throw scenario. It begins by establishing the probability of scoring a free throw as 70%, or 0.7 as a decimal, and consequently, the probability of missing as 30%, or 0.3. The focus then shifts to determining the probability of scoring exactly two free throws out of six attempts. The method involves considering the different sequences in which two scores can occur within six attempts and calculating the probability for each sequence. The probability for any specific sequence is calculated by multiplying the individual probabilities of scoring and missing in that sequence, resulting in 0.7 squared times 0.3 to the fourth power. The paragraph encourages viewers to pause and consider the problem before continuing.
📚 Applying Combinatorics to Determine the Number of Ways to Score Exactly Twice in Six Attempts
This paragraph delves into the combinatorial aspect of the problem, explaining how to calculate the number of ways to achieve exactly two scores in six free throw attempts. It introduces the concept of 'six choose two', which is a binomial coefficient notation representing the number of combinations. The formula for combinations is applied, involving factorials, to arrive at the number of different ways two scores can be achieved. The calculation simplifies to 15, indicating there are 15 distinct ways to score two out of six attempts. The probability for each of these ways is revisited, and it's multiplied by the number of combinations (15) to find the overall probability of exactly two scores in six attempts. The final calculation results in a probability of approximately 5.9535%, or roughly a 6% chance, when rounded to the nearest percentage.
🤔 Reflecting on the Likelihood of Scoring Exactly Two Free Throws
The final paragraph reflects on the outcome of the probability calculation, noting that achieving exactly two scores in six attempts is quite unlikely given a high free throw percentage of 70%. It emphasizes the improbability of this specific outcome, suggesting that with such a high success rate, it's more expected to score more than two out of six attempts. This wraps up the video script by highlighting the practical implications of the calculated probability in the context of basketball free throws.
Mindmap
Keywords
💡Probability
💡Free Throw
💡Decimal
💡Percentage
💡Attempt
💡Score
💡Miss
💡Combinatorics
💡Binomial Coefficient
💡Factorial
💡Calculation
Highlights
The probability of scoring a free throw is given as 70% or 0.7 in decimal form.
The probability of missing a free throw is calculated as 100% minus the probability of scoring, resulting in 30% or 0.3.
The concept that the sum of probabilities for all possible outcomes must equal 100% or 1 is explained.
A scenario of six free throw attempts is introduced to calculate the probability of exactly two scores.
The method to calculate the probability of a specific sequence of scores and misses in six attempts is demonstrated.
The multiplication of individual probabilities for a sequence of events is used to find the probability of that exact sequence.
Different sequences that result in exactly two scores in six attempts are considered.
The probability for any particular way of getting two scores is shown to be the same due to the multiplication of probabilities.
The total number of ways to get exactly two scores in six attempts is calculated using combinatorics.
The concept of 'six choose two' from combinatorics is used to find the number of combinations.
The formula for combinations and its application to calculate the number of ways to get two scores in six attempts is detailed.
The calculation of 'six choose two' results in 15 different ways to achieve two scores in six attempts.
The overall probability of exactly two scores in six attempts is calculated by multiplying the number of ways by the probability of each way.
The final probability calculation involves multiplying 15 (the number of ways) by 0.7 squared and 0.3 to the fourth power.
The numerical result of the probability calculation is approximately 0.05935 or about 6%.
The conclusion emphasizes that achieving exactly two scores in six attempts is fairly unlikely given a high free throw percentage.
Transcripts
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