## Maths on Tour: Dolphin Watching and Probability

My wife and I just got back from a great holiday in the Maldives – a perfect mix of luxury, relaxation and interesting activities. One of the activities we decided to invest in was a 90 minute dolphin tour.

At the time of booking it, I have to say that I was skeptical about paying \$35 each for a tour in which we might not actually see any dolphins. Having said this, we took the plunge simply because the boat ride during  sunset seemed partially worth the money.

Funnily enough before the tour started, a guide said there was an 80% chance of seeing dolphins. What a great statistic to discuss in class. Where did he pluck this statistic from? How much experience did he have as a guide? Would the chances change in different weather conditions? (Surely they’re less likely to come to the surface in bad weather?)

Anyhow, half way through the tour it became apparent that we probably wouldn’t see any dolphins. The guide thus decided that we could either take a discount or come back for another tour, free of charge.

Question: What would be the probability of seeing dolphins if you went on two tours?

We can easily calculate this using a tree diagram: D – seeing dolphins

So the probability of seeing dolphins both times would be 64% and the probabiliy of seeing dolphins at least once is 64%+16%+16& = 96%

However, students should be able to see that the use of a tree diagram here is only helpful before the first trip – not after. Since seeing dolphins on one occasion is independent of seeing them on another occasion, the probability of seeing dolphins the next time is still 80%. This is a nice example of conditional probability in which P(seeing dolphins at least once)=96% but P(seeing dolphins on the second trip given that we haven’t seen dolphins on the first trip) = 80%