math

• Probability of a simple event

  A ratio that compares the number of favorable outcomes to the number of possible outcomes
• Simple event


  An event that involves a single element of the sample space
• Probability


  • A number between 0 and 1
  • The closer the probability to 1, the more likely the event
  • The closer the probability to 0, the less likely the event
  • Probability of an impossible event is 0
  • Probability of a certain event is 1
  • If the probability of an event E is p, the probability of the complement of E is 1-p
• Probability of getting a head when tossing a coin

  • 1/2
• Probability of rolling a prime number on a number cube

  • 1/2
• Probability of drawing a diamond from a standard 52-card deck


  • 1/4
• Probability of drawing a black card from a standard 52-card deck

  • 1/13
• Probability of getting 3 heads when tossing 3 coins

  • 1/8
• Probability of getting at least 2 heads when tossing 3 coins

  • 1/2
• Probability of getting at most 2 tails when tossing 3 coins

  • 7/8
• A bag contains 7 white balls and 11 orange balls. The probability of drawing a green ball is 0 since there are no green balls.
• Probability of drawing a white ball from the bag

  • 7/18
• Probability of drawing a non-white ball from the bag

  • 11/18
• Probability of drawing a red ball after 12 red balls are added to the bag

  • 12/30
• Probability of the complement of rolling a 4 on a die

  • 5/6
• Probability of the complement of selecting a vowel from the alphabet

  • 21/26
• Probability of the complement of selecting a month starting with J

  • 3/4
• The product of two numbers is the result obtained when they are multiplied.
• A factor is one of the numbers that make up another number or expression.
• To find the factors, we need to divide the given number into pairs of equal parts.
• Triangle inequality theorem


  The sum of the lengths of any two sides of a triangle is greater than the length of the third side
• Applying triangle inequality theorem

  1. Identify pair of two sides

  2. Compare sum of two sides to third side

  3. If sum of two sides is greater than third side, then it can form a valid triangle
• Possible side lengths of a triangle

  • Set A: 7, 5, 4 (valid)
  • Set B: 9, 11, 2 (invalid)
  • Set C: 12, 10, 12 (valid)
• Finding range of possible third side length

  1. Let the given side lengths be a and b

  2. The third side length x must satisfy: a - b < x < a + b
• Finding range of possible third side length

  • Given sides: 15 cm, 9 cm

  Range of third side: 7 cm < x < 23 cm
• Finding possible values of variable x in triangle sides

  Let the sides be: 18, 2x, x

  Apply triangle inequality:

  2x - x < 18 < 2x + x

  x > 6 and x < 18

  Therefore, possible values of x: 7 ≤ x ≤ 17
• is 7 is true among the given sides of the triangle so therefore if the value of x is 7 or 7 8 9 10 up to 17 that is true among the lengths of the sides of the triangle
• six is not included in the possible values of x
• 2x plus x is greater than 18


  Substitute nothing is six so that is twelve plus six as you can see this is false so that's why six is not included for the possible values of x
• 8 x plus 18

  Any two sides x plus 18 is greater than 2x so this is 18 plus 18 substitute nothing let's say 18 so that is 36 is greater than 36 as you can see they are equal so it's not greater than
• Find the values of x given the sides of 16 x and 3x

  1. 3x minus x and then your third side sixteen and then three x plus x

  2. 3x minus x is less than sixteen

  3. 3x plus x is greater than sixteen
• The possible values of x are greater than four but less than eight
• Any pair of two sides

  • 3x plus x should be greater than 16
  • 16 plus 3x should be greater than x
  • 16 plus x is greater than 3x
• 5 to 7 are true for the possible values of x for us to form a triangle
• 4 and 8 are not part of the possible values of x as they did not form a triangle
• The sum of the two sides of a triangle should be greater than the third side
• Angle-side relationship theorem


  • The largest angle is opposite the longest side
  • The smallest angle is opposite the shortest side
• Identifying the largest angle, smallest angle, longest side, and shortest side

  • Largest angle: angle A
  • Smallest angle: angle B
  • Longest side: BC
  • Shortest side: AC
• The largest angle is always opposite the longest side, and the smallest angle is always opposite the shortest side
• Arranging the angles from largest to smallest

  1. Angle A (largest)

  2. Angle C

  3. Angle B (smallest)