Question

A box contains 8 black beads and 12 white beads. Another box contains 9 black beads and 6 white beads. One bead from each box is taken.
a) What is the probability that both beads are black?
b) What is the probability of getting one black bead and one white bead?

Answer 1

 HINT: The formula for evaluating probability of any event is
P \[=\dfrac{Favorable\ outcomes}{Total\ outcomes}\] .
Another important formula is that, For calculating number of favorable outcomes for two different events happening together, we just need to multiply the number of their respective happening chances , for example if the chances of happening an event is M and that for another different event is N, then the favorable chances of these two to happen together is \[M\times N\].

Complete step by step answer:
Now, in the question it is mentioned that there are 8 black beads and 12 white beads in one box and 9 black beads and 6 white beads in the other box. So, the total outcomes for the event of drawing beads at random from the boxes is
Total outcomes
 \[\begin{align}
  & =(8+12)\times (9+6) \\
 & =20\times 15 \\
 & =300 \\
\end{align}\]
Total outcomes =300.
Now, for favorable outcomes for part (a), we get
Favorable outcomes for part (a)
\[\begin{align}
  & =8\times 9 \\
 & =72 \\
\end{align}\]
Favorable outcomes \[=72\] .
Now, using the formula for calculating the probability of getting both beads as black is\[\begin{align}
  & =\dfrac{Favorable\ outcomes}{Total\ outcomes} \\
 & =\dfrac{72}{300} \\
 & =\dfrac{6}{25} \\
\end{align}\]
Hence, the probability of getting a red ball from the box is \[\dfrac{6}{25}\] .

Favorable outcomes for part (b)
\[\begin{align}
  & =8\times 6+9\times 12 \\
 & =48+108 \\
 & =156 \\
\end{align}\]
Favorable outcomes \[=156\] .
Now, using the formula for calculating the probability of getting one black bead and one white bead is \[\begin{align}
  & =\dfrac{Favorable\ outcomes}{Total\ outcomes} \\
 & =\dfrac{156}{300} \\
 & =\dfrac{13}{25} \\
\end{align}\]
Hence, the probability of getting a red ball from the box is \[\dfrac{13}{25}\] .

NOTE:The students can make an error if they don’t have any idea regarding the property which is for calculating number of favorable outcomes for two different events happening together, we just need to multiply the number of their respective happening chances , for example if the chances of happening an event is M and that for another different event is N, then the favorable chances of these two to happen together is \[M\times N\] .