Question

1 - 81. your team is in charge of games at the cpm amusement park. one of the games involves a robotic arm that randomly grabs a stuffed animal out of a large bin. before it grabs a stuffed animal the bin shakes to ensure all the stuffed animals are well mixed. you need to set up the game so that the probability of a customers grabbing a teddy bear is exactly 1/2. a. how would you set up the bin? explain. b. what if you returned to check on the bin and found that there were 4 teddy bears left and 12 other animals? what could you add to or remove from the bin to return the probability of selecting a teddy bear to 1/2?

Explanation:

Step1: Recall probability formula

The probability of an event $P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. We want $P(\text{selecting teddy bear})=\frac{1}{2}$.

Step2: Let the number of teddy - bears be $x$ and total number of stuffed animals be $y$.

We know $P=\frac{x}{y}=\frac{1}{2}$, so $y = 2x$.

Step3: For part a

Let the number of teddy - bears be $t$ and non - teddy stuffed animals be $n$. We want $\frac{t}{t + n}=\frac{1}{2}$. So we need to make sure that the number of teddy bears is equal to the number of non - teddy stuffed animals in the bin. For example, if we have 10 teddy bears, we should have 10 non - teddy stuffed animals.

Step4: For part b

Initially, assume the number of teddy bears is $t_1=4$ and non - teddy stuffed animals is $n_1 = 12$. The total number of stuffed animals is $T_1=4 + 12=16$, and the probability of selecting a teddy bear is $P_1=\frac{4}{16}=\frac{1}{4}$.
To make the probability $\frac{1}{2}$, let the number of teddy bears to be added be $a$ and non - teddy stuffed animals to be removed be $r$.
We want $\frac{4 + a}{16 + a-r}=\frac{1}{2}$. Cross - multiplying gives $2(4 + a)=16 + a-r$. Expanding, we get $8+2a=16 + a-r$. Rearranging, $2a - a+r=16 - 8$, so $a + r = 8$.
We could add 4 teddy bears and remove 4 non - teddy stuffed animals. Another option is to add 8 teddy bears and not remove any non - teddy stuffed animals, or remove 8 non - teddy stuffed animals and not add any teddy bears.

Answer:

a. Make the number of teddy bears equal to the number of non - teddy stuffed animals in the bin.
b. Options include adding 4 teddy bears and removing 4 non - teddy stuffed animals, adding 8 teddy bears and removing 0 non - teddy stuffed animals, or removing 8 non - teddy stuffed animals and adding 0 teddy bears.