Two teams, the Exponents and the Radicals, square off in a best of 5 math hockey tournament. Once a team wins 3 games, the tournament is over.
The schedule of the tournament (for home games) goes: E-R-E-R-E
If the Exponents are playing at home, there is a 60% chance they'll win. If they are playing on the road, there is a 45% chance they'll win.
Find the probability that the Exponents win the series. Find this probablitliy by using a tree diagram.
hard to draw trees here, will do it in listing form
we want E to win, as soon as they win 2 games it is all over, no more than 3 games
Possible ways:
EE = .6^2 = .36
ERE = .6(.4)(.6) = .144
REE = .144
prob = sum of those = .648
To find the probability that the Exponents win the series, we can use a tree diagram. The tree diagram will help us visualize all the possible outcomes and calculate the probability of each outcome.
Let's break down the schedule and possible outcomes using a tree diagram:
E
/ \
/ \
R E
/ \ / \
/ \ / \
E R E
/ \ / \ / \
/ \ / \ / \
R E R E
In each game, there are two possible outcomes: Exponents win (W) or Exponents lose (L). Since the Exponents need to win 3 out of 5 games, we can label the outcomes as WW, WL, LW, LL, and WW.
Now, let's calculate the probabilities of each outcome:
For the first game, the Exponents are playing at home, so the probability of winning is 60% or 0.6. The probability of losing is 1 - 0.6 = 0.4.
For the second game, the Exponents are playing on the road, so the probability of winning is 45% or 0.45. The probability of losing is 1 - 0.45 = 0.55.
We can now fill in the probabilities on the tree diagram:
E(0.6)
/ \
/ \
R E(0.45)
/ \ / \
/ \ / \
E(0.6) R(0.55) E(0.45)
/ \ / \ / \
/ \ / \ / \
R(0.45) E(0.6) R(0.55) E(0.45)
Calculate the probabilities of each outcome by multiplying the probabilities along the branches. For example, the probability of WW is 0.6 * 0.45 * 0.6 = 0.162.
Continue calculating the probabilities for each outcome:
WW: 0.6 * 0.45 * 0.6 = 0.162
WL: 0.6 * 0.45 * 0.55 = 0.1485
LW: 0.4 * 0.55 * 0.6 = 0.132
LL: 0.4 * 0.55 * 0.55 = 0.121
WW: 0.4 * 0.45 * 0.45 = 0.081
Now, sum up the probabilities of the favorable outcomes (WW, WL, LW) to find the probability that the Exponents win the series:
Probability of Exponents winning = WW + WL + LW = 0.162 + 0.1485 + 0.132 = 0.4425
Therefore, the probability that the Exponents win the series is 0.4425 or 44.25%.