Expected Value

Consider the lucky wheel as shown. The angle of the sector of the red region is of the round angle. If we spin 4 times, we expect the pointer to stop 3 times on the red region and 1 time on the blue region. If we spin times, we expect the pointer to stop ____ times on the red region and ____ times on the blue region. 40 8 30 6 10 2

Region where pointer stops Red (2 points) Blue (8 points) Expected frequency 30 10 On average, the points which a player gets in one spin  Total points which the player gets in 40 spins  Total number of spins 3.5 is called the expected value of points obtained in one spin of the lucky wheel. Let’s consider the step involved in the above calculation.

The expected value of points which the player gets in one spin The theoretical probabilities of getting 2 points and 8 points are and respectively. 3 4 4 1 Here, we can see that, the expected value can be calculated using probabilities of the events.

In general, Consider an activity which has n possible outcomes, and the values obtained from the possible outcomes are x1, x2, x3, …, xn. If the probabilities of the occurrence of the above possible outcomes are p1, p2, p3, …, pn respectively, then the expected value of this activity is x1p1 + x2p2 + x3p3 + … + xnpn.

Follow-up question A ball is drawn at random from 4 identical balls marked with 11, 15, 17 and 21 respectively. Find the expected value of the marks of the ball drawn. Solution ∵ The 4 balls are equally likely to be drawn. ∴ The theoretical probabilities of getting the 4 balls are all . ∴ Expected value of the marks of the ball drawn