In a game a player rolls a fair six sided die if the score...

In a game a player rolls a fair six-sided die. If the score is even, the player receives an amount of dollars equal to the score. If the score is odd, the player receives an amount of dollars equal to the score multiplied by three. It costs the player $5 to play the game. (a) What is the probability mass function of the player's net winnings? (b) What are the expectation and the standard deviation of the net winnings? (c) Suppose that a player plays the game 10 times. What are the expectation and the standard deviation of the total net winnings from all 10 games?

Transcript

00:01 Now in this question, let's look at again, you roll a death, sorry, a die.

00:07 That's a failed six -sided die.

00:09 So you have six numbers, right? one, two, three, four, five, six, right? so if you roll a die, and you may get a score, which are based on the numbers on the side of the bad, right? so you have six possible scores.

00:27 Now for each score, you're likely to get a award, right? so suppose this is the word.

00:39 Now, if you get an even number, you just get a award that amounts to the even number, right? so for two, you get $2, $4, and $6.

00:51 But if it's odd, you multiply three.

00:54 So this is $3, $9, and $15, right? now the winnings, so how much is your winnings? so the winnings obviously should be, you know, the five, three minus five, right? so in this case, it's because the cost is five, it takes you five points, it takes you five dollars to start the game.

01:19 So if you grow the diet to get a score one, and then you get $3 for reward, and you lose $2, right? so it's minus two.

01:30 And this case in minus three, right? and this is you gain four, and this is minus one.

01:36 And here you gain 10.

01:37 Here you gain 1, right? so the probability i'm going to call the winnings x, right? so i give it a symbol x.

01:45 So x denotes the waymence, right? you have six possible waynance as listed here, right? and the probability p of x obviously is 1 of 6, 1 over 6, 1 over 6, 1 over 6.

02:00 1 over 6, 1 over 6, right? because it's a failed 6 -7 diet.

02:07 So the probability for any of this month for winness is 1 of 6.

02:14 And, of course, the probability for any other x value is 0, 0 ,0 .0.

02:20 So that's the probability must function.

02:24 What are the expectation and standard variation of the net venance? winnings, right? the expectation i call it ex, right? and that, of course, is given by, that of course is given by just multiply this two columns, right? multiply this, for example, minus two times one of six and minus three times one of six, right? so you basically would get one over six times minus two plus minus three, plus four, plus minus one, plus ten, plus one, right? so that's your expected winning.

03:02 So that's given by 1 of 6 and times minus 5, minus 1, minus 2, 8, 9, so it times 9...

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