Question

I tried my best to complete these study guides but to no avail. I spent two hours looking at videos and my notes and my professors notes but no luck. Please assist. Please provide step by step set up to help me understand and not just the answers.

1. The odds in favor of event E is given as 3:17. Find (a) (2 points) P(E) 34-1 - (b) (2 points) P(E) PCE) 2. (3 points) The odds of winning a fair game is given as 1:4. How much does a player have to bet in order to have a net gain of $30? 2 3. (2 points) The probability that LA Lakers win the NBA championship this year is 0.08. Find the odds against the LA Lakers winning the NBA championship this year. 3. 4. (2 points) The odds of a winning a game is given as 1:5. What is the probability of winning the game? 4.

Answer 1

Some Important Theory.

Probability = Favourable outcomes/Total Outcomes

Sum of Probabilities of any event = 1 Therefore if p is the probability of an event happening, then 1 - p is the probability of the event not happening. We also write this as P(A) + P(A') = 1, where P(A) is the probability of the event happening and P(A') is the probability of the same event not happening.

Odds in favour of an event = p/(1-p) and odds against an event = 1-p/p

Remember, odds in favour and odds against are ratios and not probabilities. We use them to calculate probabilities.

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1) The odds in favour of an event E = 3 : 17

(a) to find P(E). Let p be = P(E) and 1 - p = P( $\bar{E}$ )

Odds in favour = 3/17 = p/(1-p).

Solving by cross multiplication , 17p = 3(1-p) or 17p = 3 - 3p. Therefore 20p = 3or p = 3/20

Therefore P(E) = 3/20

(b) Since P(E) + P( $\bar{E}$ ) = 1, Therefore P( $\bar{E}$ ) = 1 - P?(E) = 1 - 3/20 = 17/20

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2) Odds of winning a fair game = 1 : 4, which means if he invest $x he will win $4x

Now, P(Winning)/P(Not Winning) = p/(1 - p) = 1/4

Solving, 4p = 1 - p. Therefore 5p = 1 or p = 1/5 and 1 - p = 1 - 1/5 = 4/5

Let him invest $x. If he wins his net investment = 4x - x = $3x

If he loses, his net loss = -$x

Therefore Winning amount * P(Winning) + Losing Amount * P(not winning) = $30 (Net earnings)

3x * (1/4) - x * (3/4) = 30. Solving we get, 3x - x = 30 * 4 = 120

Therefore 2x = 120 or x = $60

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3) P(LA Lakers win) = 0.08, therefore P( LA Lakers lose) = 1 - 0.08 = 0.92

Odds against = (1 - p)/p = 0.92/0.08 = 92/8 = 92 : 8 = 23 : 2

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4) Odds of winning = 1 : 5. Therefore p/(1-p) = 1/5

Solving, 5p = 1 - p or 6p = 1. Therefore p = 1/6

P(Winning) = 1/6

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Theory on Coins. When flippin 1 coin, the total outcomes = 2 Heads or Tails. If you flip 2 coins, we have 4 outcomes (HH, HT, TH, TT) = 4 outcomes. So if you flip n coins, the total outcomes = 2ⁿ.

Probability = Favourable outcomes/Total Outcomes

When we flip 4 coins, the total outcomes = 2⁴ = 16

(a) P(All tails in 4 flips) = TTTT = 1 outcome (In no other outcome can we get all 4 tails)

Therefore the probability = 1/16 = 0.0625

(b) P(All heads in 4 flips) = HHHH = 1 outcome (In no other outcome can we get all 4 Heads)

Therefore the probability = 1/16 = 0.0625

(c) P(All are different). This means we do not want outcomes where all are heads and all are tails. Therefore out of the 16 outcomes, only 2 outcomes are there where all come same, i.e all heads and all tails. So 14 outcomes, we will get a mixture of heads and tails.

Probability = 14/16 = 7/8 = 0.875

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