Or search by topic
Work out the probability that, if I have an apple on Monday, I have:
Calculate ${\bf M}^2$. What do you notice about the first column of your matrix?
Tanish, Mateusz and Nishad calculated ${\bf M}^2$ and all noticed the same thing. This is Tanish's wok:
Ci Hui from Queensland Academy for Science Mathematics and Technology in Australia showed how to use the transition matrix to get the probabilities of eating an apple and a banana on Wednesday - and therefore showed why these can come from ${\bf M}^2:$
Calculate ${\bf M}^3$ and ${\bf M}^4$. What do you think will happen to ${\bf M}^n$ as $n \to \infty$?
Maiwand from LAE Tottenham in the UK, Tanish, Mateusz and Ci Hui had similar answers. This is Maiwand's work:
To decide what happens to $\bf M^n$ as $n \rightarrow \infty$, Ci Hui plotted graphs (using Excel) showing how the four elements of $\bf M^n$ change for different values of $n$ (click on the image to open a larger version)
Nishad used proof by induction to prove what happens to $\bf M^n$ as $n \rightarrow \infty.$ Click to see Nishad's proof.
On average, what proportion of the days will I choose an apple?
Tanish, Mateusz, Nishad and Ci Hui answered this correctly. This is Nishad's work:
Now, instead of just apples and bananas I can choose a cake.
Write down a transition matrix for this situation.
Tanish, Mateusz, Ci Hui and Nishad wrote down the matrix, and Bethany from England explained how to. Click to see Bethany's work:
Most people put the Cake in the third column and third row, so got a slightly different transition matrix, $\begin{pmatrix} 0 & 0.3 & 0.5 \\ 0.2 & 0.4 & 0 \\ 0.8 & 0.3 & 0.5 \end{pmatrix}.$
Use your matrix to work out the probabilities that I have an apple, banana or cake on Friday given that I had an apple on Monday.
Bethany used repeated matrix multiplication to find the probabilities:
Tanish, Mateusz and Nishad used $\bf M^4.$ Tanish wrote:
On average, what proportion of the days will I eat cake?
Tanish, Mateusz, Nishad and Ci Hui all agreed about the answer. This is Mateusz's work, which shows how they got there (click on the image to open a larger version):