# Spikes and grains

### The problem

Next seeds was accumulated to sow:

• 1 grade 95%
• 2 grade 3%
• 3 grade 2%

The probability that 1 spike has 50 seeds is:

• 0,5 for the 1-st grade
• 0,2 for 2-nd grade
• 0,1 for 3-rd grade

What the probability that one random spike has 50 seeds?

### Solution

The solution can be found by law (or formula) of total probability:

$\\ P(A) = \sum_{i=1}^{n} P(H_{i})\times P(A|H_{i})$

Along with given formula we have next solution:

$$\\ A - \texttt{is an event, one spike was grabbed from the field} \\ H_{1} - \texttt{is an event, 1-st grade spike was grabbed from the field} \\ H_{2} - \texttt{is an event, 2-nd grade spike was grabbed} \\ H_{3} - \texttt{is an event, 3-rd grade spike was grabbed} \\ P(H_{1}) = 0.5 \\ P(H_{2}) = 0.2 \\ P(H_{3}) = 0.1 \\ P(A|H_{1}) = 0.95 \\ P(A|H_{2}) = 0.03 \\ P(A|H_{3}) = 0.02 \\ P(A) = 0.5\times 0.95 + 0.2 \times 0.03 + 0.1 \times 0.02 = 0.483$$