# AS91267 Probability Exam

Could you please show full working for the second part of the question. I’ve attached the first question too

This one as well, I found it quite confusing

Hi @izzywizzy! I assume you mean part (a)(iii). We do this in two steps: first we find the proportion of unsafe rivers in each land type (from table 1), and then we use that to find the total proportion of unsafe rivers. We need to do this because we are told that the proportion of rivers in each land type is different to those in table 1, while the proportion of safe rivers in each land type can be assumed to stay the same.

We can use table 1 from (a)(i) and (ii) to estimate the proportion of unsafe rivers in each category of river.
P(unsafe | native) = 48/194 = 0.2474.
P(unsafe | exotic) = 11/26 = 0.4231
P(unsafe | pasture) = 424/528 = 0.8030
P(unsafe | urban) = 57/62 = 0.9194

Next we can use the proportion of rivers in each land type to find the total proportion of unsafe rivers. We multiply the proportion of each land type, by the probability of a river being unsafe given it is in that land type, and add all these together.
0.48*(48/194) + 0.05*(11/26) + 0.46*(424/528) + 0.01*(57/62) =
0.5185

The question asks for the ‘percentage’ so we multiply this by 100 to get our final answer.
0.5185*100 = 51.85%
According to this data, 51.85% of all New Zealand rivers are unsafe for swimming.

So that’s part (iii) out of the way! Now we can take a look at (b)(ii).

In part (b)(i) you (hopefully) found that when sd = 450L and P(x<2000) = 0.85 the mean would be 1533.6L.

For such a distribution, P(x>15,000) is miniscule - essentially zero. Additionally, when P(x>X) = 0.015, X = 2000L (if this distribution was applicable, 1.5% of businesses would use more than only 2000L a day).
If 1.5% of businesses actually use more than 15,000L a day, this suggests that the mean found in part (i) is incorrect. Moreover, it suggests that the distribution of water usage for Auckland businesses might not even be a normal distribution. This makes sense as we know that water usage cannot be less than zero, so there will be a lower limit, but there is essentially no upper limit - causing a tail to the right. A lot (85%) of businesses will use <2000L of water/day but there are a small amount (1.5%) of businesses, such as pools, factories or farms which may use a very large amount of water, >15,000L/day. This suggests that a normal distribution would not accurately model this situation.

For the question highlighted, I discussed that I am not confident that this probability would be correct for all NZ rivers because every region experiences different weather patterns and there is always variation in terms of climate eg. wind/areas that receive greater precipitation than others. As such, these environmental factors can affect the quality of water and whether they are safe/unsafe for swimming.

Is my reasoning above valid? If not, what’s the correct answer…I assumed that we were supposed to come up with a statistical reason lol

That could definitely be a valid reason.

You could also discuss sample size and sampling method, as we are only told that “some rivers” were tested, not how they were selected. The sample of rivers from exotic forest or urban rivers is also not very big.