Leia is operating a digital spinner game in which a player spins an arrow and is given a prize based on where the spinner lands. Leia can program the probability of each of the outcomes. The stuffed teddy bear is the best prize, so she programs the spinner so that the probability of not getting a bear twice in a row is greater than 3 times the probability of getting the teddy bear in one spin.

Write an inequality that compares the probability of getting the teddy bear to the probability of not getting the teddy bear, using p to represent the probability of getting the teddy bear in one try.

In the problem, we are given that the probability of getting a bear is equal to p. Where probability is the share of successful outcome in this case is the bear) over total outcomes (total amount of prizes).

The likelihood of not receiving a bear two times in a row is containing of 2 tries:

try 1: 1st spin gets a bear, 2nd spin does not receive a bear

total probability of try 1 = p * (1 - p)

try 2: 1st spin does not get a bear, 2nd spin gets a bear

total probability of try 1 = (1 - p) * p

The total probability of the 2 tries is just the sum of the 2:

Overall probability of not getting a bear twice in a row = p * (1 - p) + (1 - p) * p

Simplifying this will give us:

Overall probability of not getting a bear twice in a row = 2p * (1 - p)

Therefore, the likelihood of not receiving a bear twice in a row > 3 times the probability of getting the teddy bear. Rewriting the statement, will give us the inequality of 2p * (1 - p) > 3 p