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2013 Question 3c

Hi there!
I am really struggling on this problem and would very much appreciate any help I can get. So please do help me!

In 2013 Question 3c. in Probability Distributions, I cannot understand the answer. For example, could you please explain to me why they calculate the probability is greater than 13? How would you know to do this? Why cannot we calculate p(x=13) and compare that to the actual result. Is it possible if you could just go through the whole process of answering that particular question.

I am aiming for excellence and usually I do not find calculation questions that difficult. I just really hope that I can understand this question properly. Sorry if this is a dumb question.

Hi there
There are no dumb questions Newton!

First thing to do is identify the type of distribution that suits this situation. Two possible answers (it lands logo up or logo down), there are a fixed number of trials (it was spun 20 times and the position of the logo was noted), each trial is independent (how it lands each time has no impact on how it will land another time) and the same chance of a success (presume that there is a half chance of it landing each way). This leads us to Binomial distribution. Make sure you justify this by going through each condition.
Next look at the specific case. We are wondering if getting 13 is likely due to random variability or not. We look at 13 or more landing facing up to see if that “tail” of possible results is likely with a probability value of 0.5. If 13 or more has a really small probability of occurring, based on sampling variability, then there is evidence to suggest that 13 times of landing logo up is unlikely to be a random event and more likely attributed to an unbalanced racket. In this case, with a probability of 0.1316 (or 13.16%), this isn’t particularly small and therefore it is likely that this number of times that the racket landed logo up could be attributed to random variability rather than unbalanced rackets. This is also backed up by having a small number of trials (only 20) and therefore it is actually very unlikely to get exactly half of the spins for the racket to land logo side up. Remember that random variability means that every time you run an experiment or take a sample, you will get different results.

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Thank you.

Can you please check what I am saying is correct because I am going to try to put it into my own words

So essentially, since the probability of it landing more than 13 times is not small, it suggests that the probability of it landing up is not equal to the probability of it landing down. However, this may be due to random variability as there can be variability with the results i.e. we expect variation in the results. Furthermore, since this is just a sample of 20, it is not a big enough sample so we have to consider random variability. There should be more trials and a bigger sample to account for sampling variation, and then we can come to a decision on whether the probabilities of it landing up = landing down.

Once again thanks a lot. You have been a big help!

Kia ora @NewtonsApple

I think you are misinterpreting what MinEdSupport has said. The fact that the probability of it landing up 13 times or more using the Binomial distribution is 13.16% shows that this is likely to have happened from chance alone. Sometimes it is useful to look at the binomial graph to see if it makes sense (see below). It is not suggesting that the probability of landing up is not equal to that of it landing down.
Had it have landed say 18 times on the logo there may have been some evidence, but you are correct that using only a sample of 20 is not really big enough.
binomial

Hi @Algebra123
So essentially, since the probability of it landing more than 13 times is not that small, it must be because of random variability? If it was very small, then it is likely that the rackets are unbalanced. Is that right?

I really appreciate your help. Thanks

Yes @NewtonsApple the binomial calculation indicates that it is quite possible for it to land 13 or more times. A key thing here is that the experiment is only 20 tosses, with a larger experiment you would be more likely to get closer to theoretical probability. To further investigate if the racket was actually unbalanced I would suggest more testing.

Happy to help :smile:

@Algebra123 I think I understand it better now, and if I do not I will make sure to ask you again. I would just like to thank you for the very clear explanation and the time you took to make the graph for me! Very much appreciated!

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