Answer: 0.420175
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Explanation:
Let's consider the case where Ed makes all 6 shots. The probability of this happening is (7/10)^6 = 0.117649
Let A = 0.117649 so we can use it later.
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Now consider the case where Ed makes exactly 5 shots. So he misses exactly 1 shot. Furthermore, consider the case where his first shot is a miss and the remaining 5 shots are successful.
The probability of getting the first shot a miss and the remaining 5 successful is (3/10)*(7/10)^5 = 0.050421
Note how 3/10 and 7/10 add to 1. The 3/10 is the probability of getting a miss (since he makes 7 goals for every 10 shots, he misses those 3 goals).
The value 0.050421 represents the probability for that first shot being a failed attempt while the rest are successful shots. Since there are 6 slots total, this means 6*0.050421 = 0.302526 represents the probability of missing exactly one goal and hitting the others. By this point, we aren't making the failed attempt at slot 1. It could be at any slot. For instance, the failed attempt could be at slot 4.
In other words, 0.302526 is the probability of making exactly 5 shots.
Let B = 0.302526
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To recap,
- A = probability of getting all 6 shots in the goal (no failed attempts).
- B = probability of getting exactly 5 shots in the goal (exactly one attempt failed).
These two events are mutually exclusive. They cannot happen at the same time. That allows us to add the values of A and B to get the final answer
A+B = 0.117649+0.302526 = 0.420175
Therefore, A+B = 0.420175 represents the probability of either getting all 6 shots in the goal OR getting exactly five shots in the goal.
In short, 0.420175 represents the probability of making at least 5 shots. The phrasing "at least 5" means "5 or more".
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If you wanted to calculate this in one line, then you could say
(7/10)^6 + 6*(3/10)*(7/10)^5 = 0.420175
This decimal value is exact.
